#### Prof. Dr. Klaus Hackl

Mechanics

Ruhr-Universität Bochum

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**Acoustic waveform inversion in frequency domain: Application to a tunnel environment**

Riedel, C. and Musayev, K. and Baitsch, M. and Zhu, H. and Hackl, K.*Underground Space (China)*(2021)Waveform inversion is an approach used to find an optimal model for the velocity field of a ground structure such that the dynamic response is close enough to the given seismic data. First, a suitable numerical approach is employed to establish a realistic forward computer model. The forward problem is solved in the frequency domain using higher-order finite elements. The velocity field is inverted over a specific number of discrete frequencies, thereby reducing the computational cost of the forward calculation and the nonlinearity of the inverse problem. The results are presented for different frequency sets and with different source and receiver locations for a two-dimensional model. The influence of attenuation effects is also investigated. The results of two blind tests are presented where only the seismic records of an unknown synthetic model with an inhomogeneous material parameter distribution are provided to mimic a more realistic case. Finally, the result of the inversion in a three-dimensional space is illustrated. © 2021 Tongji Universityview abstract 10.1016/j.undsp.2021.01.001 **An effective model for cancellous bone with a viscous interstitial fluid ***

Blaszczyk, M. and Pertsch Gilbert, R. and Hackl, K.*Applicable Analysis*(2021)We outline the mathematical model of the ultrasonic response of cancellous bone and its time harmonic formulation. In contrast to the Biot model, the fluid is not inviscid. Our fluid is viscous, but does not interact with the solid components. © 2021 Informa UK Limited, trading as Taylor & Francis Group.view abstract 10.1080/00036811.2021.1979221 **Multiscale modeling of cancellous bone considering full coupling of mechanical, electric and magnetic effects**

Blaszczyk, M. and Hackl, K.*Biomechanics and Modeling in Mechanobiology*(2021)Modeling of cancellous bone has important applications in the detection and treatment of fatigue fractures and diseases like osteoporosis. In this paper, we present a fully coupled multiscale approach considering mechanical, electric and magnetic effects by using the multiscale finite element method and a two-phase material model on the microscale. We show numerical results for both scales, including calculations for a femur bone, comparing a healthy bone to ones affected by different stages of osteoporosis. Here, the magnetic field strength resulting from a small mechanical impact decreases drastically for later stages of the disease, confirming experimental research. © 2021, The Author(s).view abstract 10.1007/s10237-021-01525-6 **The effective equations for the ultrasonic response of wet cortical bone**

Blaszczyk, M. and Pertsch Gilbert, R. and Hackl, K.*Mathematical Methods in the Applied Sciences*44 (2021)We outline the mathematical model of the ultrasonic response of wet cortical bone and its time-harmonic formulation. We employ an energetic approach based on the Reuss bound of the free energy of a porous material consisting of a piezo-electric solid and a conducting fluid part. Magnetic effects are taken into consideration. Corresponding boundary value problems are stated, and associated theorems are established. A conclusion is included concerning future developments of this formulation. © 2021 John Wiley & Sons, Ltd.view abstract 10.1002/mma.7337 **Adaptive Concurrent Topology Optimization of Coated Structures with Nonperiodic Infill for Additive Manufacturing**

Hoang, V.-N. and Tran, P. and Nguyen, N.-L. and Hackl, K. and Nguyen-Xuan, H.*CAD Computer Aided Design*129 (2020)The present research develops a direct multiscale topology optimization method for additive manufacturing (AM) of coated structures with nonperiodic infill by employing an adaptive mapping technique of adaptive geometric components (AGCs). The AGCs consist of a framework of macro-sandwich bars that represent the macrostructure with the solid coating and a network of micro-solid bars that represent the nonperiodic infill at the microstructural scale. The macrostructure including the coating skin and the internal architecture of the microstructures of cellular structures is simultaneously optimized by straightforwardly searching optimal geometries of the AGCs. Compared with most existing methods, the proposed method does not require material homogenization technique at the microscale; the continuity of microstructures and structural porosities are ensured without additional constraints; Finite element analysis (FEA) and geometric parameter updates are required only once for each optimization iteration. AGCs allow us to model coated structures with porosity infill on a coarse finite element mesh. The adaptive mapping technique may reduce mapping time by up to 50%. Besides, it is easy to control the length scales of the coating and infill as desired to make it possible with AM. This investigation also explores the ability to realize concurrent designs of coated structures with nonperiodic infill patterns using 3D printing techniques. © 2020 Elsevier Ltdview abstract 10.1016/j.cad.2020.102918 **Are Onsager's reciprocal relations necessary to apply Thermodynamic Extremal Principles?**

Hackl, K. and Fischer, F.D. and Zickler, G.A. and Svoboda, J.*Journal of the Mechanics and Physics of Solids*135 (2020)Onsager's Reciprocal Relations between thermodynamic forces and fluxes, for which Onsager was awarded the Nobel Prize, automatically follow from Thermodynamic Extremal Principles. Thus, the Principles are up to now non-applicable for the treatment of experimentally determined or theoretically modeled non-reciprocal systems as e.g. those in the magnetic field. However, we can demonstrate that adding of a certain barrier constraint as bilinear form of thermodynamic forces and fluxes accounted by the Thermodynamic Extremal Principles provides to non-reciprocal relations between the thermodynamic forces and fluxes. Such a novel idea may contribute to a better understanding of physics behind non-reciprocal systems. © 2019 Elsevier Ltdview abstract 10.1016/j.jmps.2019.103780 **Constraints in thermodynamic extremal principles for non-local dissipative processes**

Hackl, K. and Fischer, F.D. and Svoboda, J.*Continuum Mechanics and Thermodynamics*32 (2020)Phenomena treated by non-equilibrium thermodynamics can be very effectively described by thermodynamic variational principles. The remarkable advantage of such an approach consists in possibility to account for an arbitrary number of constraints among state or kinetic variables stemming, e.g., from conservation laws or balance equations. As shown in the current paper, the variational principles can provide original evolution equations for the state variables implicitly respecting the constraints. Moreover, the variational approach allows formulating the problem directly in discrete state variables and deriving their evolution equations without the necessity to solve partial differential equations. The variational approach makes it also possible to use different kinetic variables in formulation of dissipation and dissipation function. © 2019, The Author(s).view abstract 10.1007/s00161-019-00846-3 **Forward and inverse problems in piezoelectricity using isogeometric symmetric Galerkin boundary element method and level set method**

Nguyen, B.H. and Nanthakumar, S.S. and He, Y.Q. and Tran, H.D. and Hackl, K. and Zhuang, X.*Engineering Analysis with Boundary Elements*113 (2020)This work presents the method to combine isogeometric analysis coupled to symmetric Galerkin boundary element method (IGA-SGBEM) and parametric level set (PaLS)-based optimization scheme for the analysis of linear problems in two-dimensional piezoelectric media. IGA-SGBEM is used to obtain field variables (i.e. generalized displacement and traction) in the forward analysis. Then, inverse analysis of flaw detection in piezoelectric structures is attempted by combining IGA-SGBEM with PaLS-based optimization scheme. In this proposed method, the versatility of isogeometric analysis (IGA) is proved in the inverse progress, where the iso-line of the level set function can be easily reconstructed and incorporated into the IGA framework. Numerical examples are examined to validate and to demonstrate the robustness of the proposed method in solving both forward and inverse problems in piezoelectricity. © 2019view abstract 10.1016/j.enganabound.2019.12.020 **On the treatment of non-reciprocal rate-independent kinetics via thermodynamic extremal principles**

Hackl, K. and Fischer, F.D. and Svoboda, J.*Journal of the Mechanics and Physics of Solids*145 (2020)Onsager's Reciprocal Relations between thermodynamic forces and fluxes, for which Onsager was awarded the Nobel Prize, automatically follow from Thermodynamic Extremal Principles. Thus, the principles have been up to now non-applicable for the treatment of experimentally determined or theoretically modeled non-reciprocal systems as e.g. those involving magnetic fields. Recently, we were able to demonstrate that adding of a certain barrier constraint as bilinear form of thermodynamic forces and fluxes accounted by the Thermodynamic Extremal Principles leads to non-reciprocal relations between the thermodynamic forces and fluxes. In this work, we extend this formulation to rate-independent systems possessing non-differentiable dissipation functions. As an application, we show that the non-associated models of pressure dependent plasticity can be obtained in this fashion. © 2020view abstract 10.1016/j.jmps.2020.104149 **Sensitivity analysis of non-local damage in soft biological tissues**

Zuo, D. and Avril, S. and Ran, C. and Yang, H. and Mousavi, S.J. and Hackl, K. and He, Y.*International Journal for Numerical Methods in Biomedical Engineering*(2020)Computational modeling can provide insight into understanding the damage mechanisms of soft biological tissues. Our gradient-enhanced damage model presented in a previous publication has shown advantages in considering the internal length scales and in satisfying mesh independence for simulating damage, growth and remodeling processes. Performing sensitivity analyses for this model is an essential step towards applications involving uncertain patient-specific data. In this paper, a numerical analysis approach is developed. For that we integrate two existing methods, that is, the gradient-enhanced damage model and the surrogate model-based probability analysis. To increase the computational efficiency of the Monte Carlo method in uncertainty propagation for the nonlinear hyperelastic damage analysis, the surrogate model based on Legendre polynomial series is employed to replace the direct FEM solutions, and the sparse grid collocation method (SGCM) is adopted for setting the collocation points to further reduce the computational cost in training the surrogate model. The effectiveness of the proposed approach is illustrated by two numerical examples, including an application of artery dilatation mimicking to the clinical problem of balloon angioplasty. © 2020 John Wiley & Sons, Ltd.view abstract 10.1002/cnm.3427 **Three-dimensional numerical simulation of soft-tissue wound healing using constrained-mixture anisotropic hyperelasticity and gradient-enhanced damage mechanics**

Zuo, D. and Avril, S. and Yang, H. and Jamaleddin Mousavi, S. and Hackl, K. and He, Y.*Journal of the Royal Society Interface*17 (2020)Healing of soft biological tissues is the process of self-recovery or self-repair after injury or damage to the extracellular matrix (ECM). In this work, we assume that healing is a stress-driven process, which works at recovering a homeostatic stress metric in the tissue by replacing the damaged ECM with a new undamaged one. For that, a gradient-enhanced continuum healing model is developed for three-dimensional anisotropic tissues using the modified anisotropic Holzapfel-Gasser-Ogden constitutive model. An adaptive stress-driven approach is proposed for the deposition of new collagen fibres during healing with orientations assigned depending on the principal stress direction. The intrinsic length scales of soft tissues are considered through the gradient-enhanced term, and growth and remodelling are simulated by a constrained-mixture model with temporal homogenization. The proposed model is implemented in the finite-element package Abaqus by means of a user subroutine UEL. Three numerical examples have been achieved to illustrate the performance of the proposed model in simulating the healing process with various damage situations, converging towards stress homeostasis. The orientations of newly deposited collagen fibres and the sensitivity to intrinsic length scales are studied through these examples, showing that both have a significant impact on temporal evolutions of the stress distribution and on the size of the damage region. Applications of the approach to carry out in silico experiments of wound healing are promising and show good agreement with existing experiment results. © 2020 The Author(s) Published by the Royal Society. All rights reserved.view abstract 10.1098/rsif.2019.0708 **Topology optimization with anisotropic materials, including a filter to smooth fiber pathways**

Jantos, D.R. and Hackl, K. and Junker, P.*Structural and Multidisciplinary Optimization*61 (2020)In a recent publication, an approach to optimize the orientation of anisotropic materials was presented. This strategy was embedded into the thermodynamic topology optimization based on growth. In this paper, we show that the thermodynamic orientation optimization can also be used in more classical approaches to topology optimization. We furthermore enhance the approach by a novel filtering technique to provide control over the smoothness of the pathway of principal material directions, i.e., the curvature of fibers. The filter is based on a convolution operator and is applied to the material stiffness tensor, so that the filtering technique is not directly bounded to the actual parameterization for the design variables. To this end, the topology is defined by a continuous density approach with penalization of intermediate densities (SIMP) solved via the optimality criteria method (OCM). A set of three continuous Euler angles is used as additional design variables to describe the local material rotation of the anisotropic base material. The thermodynamic optimization of the material orientation is performed by evolution of the Euler angles to minimize the elastic energy. The related evolution equations are derived by means of the Hamilton principle, well-known from material modeling. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00158-019-02461-x **Variational Modeling and Finite-Element Simulation of Functional Fatigue in Polycrystalline Shape Memory Alloys**

Waimann, J. and Hackl, K. and Junker, P.*Journal of Optimization Theory and Applications*184 (2020)Based on our previous works, we present the finite-element implementation of an energy-based material model that displays the effect of functional fatigue of shape memory alloys during cyclic loading. The functional degradation is included in our model by taking account of irreversible martensitic volume fractions. Three internal variables are used: reversible and irreversible volume fractions for the crystallographic phases and Euler angles for parametrization of the martensite strain orientation. The evolution of the volume fractions is modeled in a rate-independent manner, whereas a viscous approach is employed for the Euler angles, which account for the materials’ polycrystalline structure. For the case of a cyclically loaded wire, we calibrate our model using experimental data. The calibration serves as input for the simulation of two more complex boundary value problems to demonstrate the functionality of our material model for localized phase transformations. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.view abstract 10.1007/s10957-019-01476-0 **Variational regularization of damage models based on the emulated RVE**

Schwarz, S. and Junker, P. and Hackl, K.*Continuum Mechanics and Thermodynamics*(2020)Material models exhibiting softening effects due to damage or localization share the problem of leading to ill-posed boundary value problems that lead to physically meaningless, mesh-dependent finite element results. It is thus necessary to apply regularization techniques that couple local behavior, described, e.g., by internal variables, at a spatial level. The common way to do this is to take into account higher gradients of the field variables, thus introducing an internal length scale. In this paper, we suggest a different approach to regularization that does not make use of any nonlocal enhancement like the inclusion of higher gradients or integration over local sub-domains nor of any classical viscous effects. Instead we perform an appropriate relaxation of the (condensed) free energy in a time-incremental setting which leads to a modified energy that is coercive and satisfies quasiconvexity in an approximate way. Thus, in every time increment a regular boundary value problem is solved. The proposed approach holds the same advantage as other methods, but with less numerical effort. We start with the theoretical derivation, discuss a rate-independent version of the proposed model and present details of the numerical treatment. Finally, we give finite element results that demonstrate the efficiency of this new approach. © 2020, The Author(s).view abstract 10.1007/s00161-020-00886-0 **A canonical rate-independent model of geometrically linear isotropic gradient plasticity with isotropic hardening and plastic spin accounting for the Burgers vector**

Ebobisse, F. and Hackl, K. and Neff, P.*Continuum Mechanics and Thermodynamics*31 (2019)In this paper, we propose a canonical variational framework for rate-independent phenomenological geometrically linear gradient plasticity with plastic spin. The model combines the additive decomposition of the total distortion into non-symmetric elastic and plastic distortions, with a defect energy contribution taking account of the Burgers vector through a dependence only on the dislocation density tensor Curlp giving rise to a non-symmetric nonlocal backstress, and isotropic hardening response only depending on the accumulated equivalent plastic strain. The model is fully isotropic and satisfies linearized gauge invariance conditions, i.e., only true state variables appear. The model satisfies also the principle of maximum dissipation which allows to show existence for the weak formulation. For this result, a recently introduced Korn’s inequality for incompatible tensor fields is necessary. Uniqueness is shown in the class of strong solutions. For vanishing energetic length scale, the model reduces to classical elasto-plasticity with symmetric plastic strain εp and standard isotropic hardening. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00161-019-00755-5 **A Concept for the Estimation of Soil-Tool Abrasive Wear Using ASTM-G65 Test Data**

Hoormazdi, G. and Küpferle, J. and Röttger, A. and Theisen, W. and Hackl, K.*International Journal of Civil Engineering*17 (2019)In this paper, a new procedure for the prediction of soil-tool abrasive wear is presented which drastically reduces the duration and, therefore, the cost of simulations in comparison to conventional 3D wear modeling. The goal is to extend the experimental data from a single scratch test to the wear of mixtures by means of equations obtained from discrete element method (DEM) simulations and geometric relations. We are predicting abrasive wear with a combination of numerical and experimental approaches taking two shapes of particles into account. Single wear is quantified by measuring the width of scratch induced by a single quartz particle. Geometrical relations together with the particle’s microscopic picture are used to find the depth of scratch. DEM mixture simulations result in equations for the number of contacts and normal contact forces. Finally, the wear rate is calculated for a specific soil sample as an example to clarify the developed prediction procedure. The DEM simulations are performed using PFC 3 D code for both a homogeneous soil sample and a mixture of two different soils. We are specially investigating a relation to predict the abrasive wear caused by a mixture of particles. © 2018, Iran University of Science and Technology.view abstract 10.1007/s40999-018-0333-9 **A fast and robust numerical treatment of a gradient-enhanced model for brittle damage**

Junker, P. and Schwarz, S. and Jantos, D.R. and Hackl, K.*International Journal for Multiscale Computational Engineering*17 (2019)Damage processes are modeled by a softening behavior in a stress/strain diagram. This reveals that the stiffness loses its ellipticity and the energy is thus not coercive. A numerical implementation of such ill-posed problems yields results that are strongly dependent on the chosen spatial discretization. Consequently, regularization strategies have to be employed that render the problem well-posed. A prominent method for regularization is a gradient enhancement of the free energy. This, however, results in field equations that have to be solved in parallel to the Euler-Lagrange equation for the displacement field. An usual finite element treatment thus deals with an increased number of nodal unknowns, which remarkably increases numerical costs. We present a gradient-enhanced material model for brittle damage using Hamilton’s principle for nonconservative continua. We propose an improved algorithm, which is based on a combination of the finite element and strategies from meshless methods, for a fast update of the field function. This treatment keeps the numerical effort limited and close to purely elastic problems. Several boundary value problems prove the mesh-independence of the results. © 2019 by Begell House, Inc.view abstract 10.1615/IntJMultCompEng.2018027813 **A fully-relaxed variationally-consistent framework for inelastic micro-sphere models: Finite viscoelasticity**

Govindjee, S. and Zoller, M.J. and Hackl, K.*Journal of the Mechanics and Physics of Solids*127 (2019)The micro-sphere modeling framework provides a popular means by which one-dimensional mechanical models can easily and quickly be generalized into three-dimensional stress-strain models. The essential notion of the framework, similar to homogenization theory, is that one allows the microstructural kinematic fields to relax subject to a constraint connected to a macroscopic deformation measure. In its standard presentation, the micro-sphere modeling framework is strictly applicable to elastic materials. Presentations considering inelastic phenomena invariably, and inconsistently, assume an affine relation between inelastic macroscopic and microscopic phenomena. In this work we present a methodology by which one can lift this modeling restriction using two formally different approaches. In particular, we show how one can construct and apply a homogenization with Biot theory to generate fully-relaxed variationally-consistent macroscopic models for inelastic materials within the context of the micro-sphere model. The primary application example will be finite deformation viscoelasticity. © 2019 Elsevier Ltdview abstract 10.1016/j.jmps.2019.02.014 **Gradient-enhanced continuum models of healing in damaged soft tissues**

He, Y. and Zuo, D. and Hackl, K. and Yang, H. and Mousavi, S.J. and Avril, S.*Biomechanics and Modeling in Mechanobiology*18 (2019)Healing of soft biological tissue is the process of self-recovering or self-repairing the injured or damaged extracellular matrix (ECM). Healing is assumed to be stress-driven, with the objective of returning to a homeostatic stress metrics in the tissue after replacing the damaged ECM with new undamaged one. However, based on the existence of intrinsic length scales in soft tissues, it is thought that computational models of healing should be non-local. In the present study, we introduce for the first time two gradient-enhanced constitutive healing models for soft tissues including non-local variables. The first model combines a continuum damage model with a temporally homogenized growth model, where the growth direction is determined according to local principal stress directions. The second one is based on a gradient-enhanced healing model with continuously recoverable damage variable. Both models are implemented in the finite-element package Abaqus by means of a user subroutine UEL. Three two-dimensional situations simulating the healing process of soft tissues are modeled numerically with both models, and their application for simulation of balloon angioplasty is provided by illustrating the change of damage field and geometry in the media layer throughout the healing process. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s10237-019-01155-z **Influence of bulk energy and triple junction mobility on interface kinetics - A tool for interpretation of experiments**

Hackl, K. and Khan, A.U. and Fischer, F.D. and Svoboda, J.*Acta Materialia*174 (2019)A material system consisting of a lamellar grain structure adjacent to a large single grain is investigated. The system evolution is driven by changing of interface energy of the lamellar structure as well as by difference in bulk energies stored in the single grain and the lamellar grains. The triple junctions and the grain boundaries are assumed to have finite mobilities representing kinetic material parameters of system. A complete analysis of the kinetics of this system is provided, which involves several possible scenarios depending on the values of the geometrical and material parameters of the system. The scenarios are fully classified. Moreover, the analysis offers a way, how the values of the material parameters (interface energy densities, difference in bulk energies and mobilities) can be extracted from the measured system kinetics and geometry. © 2019 Acta Materialia Inc.view abstract 10.1016/j.actamat.2019.05.005 **Preface: Multiscale plasticity and related topics**

Hackl, K. and Schröder, J.*International Journal for Multiscale Computational Engineering*17 (2019)view abstract 10.1615/IntJMultCompEng.v17.i2.10 **A polytree-based adaptive polygonal finite element method for multi-material topology optimization**

Chau, K.N. and Chau, K.N. and Ngo, T. and Hackl, K. and Nguyen-Xuan, H.*Computer Methods in Applied Mechanics and Engineering*332 (2018)This study presents a polytree-based adaptive methodology for multi-material topology optimization (MMTOP). Polytree data structure is introduced as a general recursive multi-level mesh that is automatically refined in processing based on error analysis. In order to resolve hanging nodes in element edges, the Wachspress coordinate is employed on a reference element before using a mapping scheme to obtain shape functions and their derivatives for any polygons. A new definition of filter radius is also proposed to improve the efficiency of filters and optimized results. The combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases (including void phase), but also decreases the computing time of the overall process in comparison to using the regular fine meshes. Several benchmark and practical problems are considered to show distinct features of the proposed method. © 2017 Elsevier B.V.view abstract 10.1016/j.cma.2017.07.035 **An accurate and fast regularization approach to thermodynamic topology optimization**

Jantos, D.R. and Hackl, K. and Junker, P.*International Journal for Numerical Methods in Engineering*(2018)In a series of previous works, we established a novel approach to topology optimization for compliance minimization based on thermodynamic principles known from the field of material modeling. Hamilton's principle for dissipative processes directly yields a partial differential equation (referred to as the evolution equation) as an update scheme for the spatial distribution of density mass describing the topology. Consequently, no additional mathematical minimization algorithms are needed. In this article, we introduce a regularization scheme by penalization of the gradient of the spatial distribution of mass density. The parabolic evolution equation (owing to a similar structure to the transient heat-conduction equation) is solved most efficiently by an explicit time discretization. The Laplace operator is discretized via a Taylor series expansion yielding an operator matrix that is constant for the entire optimization process. This method shares some similarities to meshless methods and allows for an accurate application also on unstructured finite element meshes. The minimal size of the structure member can directly be controlled, a priori, by a numerical parameter introduced along with the regularization, similar to classical filter radii. © 2018 John Wiley & Sons, Ltd.view abstract 10.1002/nme.5988 **Comparison of thermodynamic topology optimization with SIMP**

Jantos, D.R. and Riedel, C. and Hackl, K. and Junker, P.*Continuum Mechanics and Thermodynamics*(2018)Computationally efficient approaches to topology optimization usually include heuristic update and/or filtering schemes to overcome numerical problems such as the well-known checkerboarding phenomenon, local minima, and the associated mesh dependency. In a series of papers, Hamilton’s principle, which originates from thermodynamic material modeling, was applied to derive a model for topology optimization based on a novel conceptual structure: utilization of this thermodynamic approach resulted in an evolution equation for the local mass distribution as the update scheme during the iterative optimization process. Although this resulted in topologies comparable to those from classical optimization schemes, no direct linkage between these different approaches has yet been drawn. In this contribution, we present a detailed comparison of the new approach to the well-established SIMP approach. To this end, minor modifications of the original thermodynamic approach yield an optimization process with a numerical efficiency that is comparable to that of SIMP approaches. However, a great advantage of the new approach arises from results that are parameter- and mesh-independent, although neither filtering techniques nor gradient constraints are applied. Several 2D and 3D examples are discussed and serve as a profound basis for an extensive comparison, which also helps to reveal similarities and differences between the individual approaches. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00161-018-0706-y **Modeling of microstructures in a cosserat continuum using relaxed energies**

Khan, M.S. and Hackl, K.*Springer INdAM Series*27 (2018)Granular materials tend to exhibit distinct patterns under deformation consisting of layers of counter-rotating particles. In this article, we are going to model this phenomenon on a continuum level by employing the calculus of variations, specifically the concept of energy relaxation. In the framework of Cosserat continuum theory the free energy of the material is enriched with an interaction energy potential taking into account the counter rotations of the particles. The total energy thus becomes non-quasiconvex, giving rise to the development of microstructures. Relaxation theory is then applied to compute its exact quasiconvex envelope. It is worth mentioning that there are no further assumptions necessary here. The computed relaxed energy yields all possible displacement and micro-rotation field fluctuations as minimizers. Based on a two-field variational principle the constitutive response of the material is derived. Results from numerical computations demonstrating the properties of relaxed potential are shown. © 2018, Springer International Publishing AG, part of Springer Nature.view abstract 10.1007/978-3-319-75940-1_6 **The effect of plasticity on damage evolution using a relaxation-based material model**

Schwarz, S. and Hackl, K. and Junker, P.*Journal of the Mechanical Behavior of Materials*27 (2018)As damage occurs in the context of high stresses that are also related to the presence of plastic strains, it is natural to investigate the effect of plasticity on damage evolution and to thus achieve a more realistic model. In this work, the existing and new damage model presented in [Junker P, Schwarz S, Makowski J, Hackl K. Continuum Mech. Therm. 2017, 29 (1), 291-310] is enhanced with plasticity and isotropic hardening. The damage model is based on a relaxation-based approach and does not require additional complex regularization techniques besides considering viscous effects. The benefit of the model are mesh-independent results for the rate-dependent case, even without considering, e.g. gradient terms for mathematical regularization. The enhancement with plasticity and isotropic hardening was investigated for a representative volume element that considerd a damaging matrix material and non-damaging hard precipitates. Two different loading types, pure tension and pure shear, yielded the homogenized stress/strain response for the material at various loading rates. Hereto, several finite discretizations in terms of finite-element meshes were used. The results underline the mesh-independence for physically reasonable loading rates and viscosities. © 2018 Walter de Gruyter GmbH, Berlin/Boston.view abstract 10.1515/jmbm-2018-2001 **A coupled computational approach for the simulation of soil excavation and transport in earth-pressure balance shield machines**

Dang, T.S. and Wessels, N. and Nguyen, N.-S. and Hackl, K. and Meschke, G.*International Journal for Multiscale Computational Engineering*15 (2017)A prototype modeling framework for the coupled simulation of excavation processes at the tunnel face and the subsequent transport of the foam-soil mixture within the pressure chamber of EPB shield machines is proposed. The discrete element method is used for the modeling of soil excavation and the stabilized finite element method, using a non-Newtonian fluid model, is employed for the modeling of fluid transport. A variational approach is applied to directly obtain interparticle parameters of the DEM from a macroscopic strength criterion. A 2D numerical simulation model for a simplified representation of the cutting process at the tunnel face and the transport of the excavated soil-foam mixture is used to demonstrate the proposed coupled excavation-transport modeling approach. According to the proposed coupled DEM-FEM model, the mass flow obtained from the excavation simulation by means of the DEM serves as the input for the finite element flow simulation to generate the pressure distribution within the excavation chamber. It is shown that the proposed approach helps to obtain insight into the coupled excavation and transport processes at the tunnel face and the spatiotemporal distribution of the face pressure. © 2017 by Begell House, Inc.view abstract 10.1615/IntJMultCompEng.2017020271 **A polytree-based adaptive approach to limit analysis of cracked structures**

Nguyen-Xuan, H. and Nguyen-Hoang, S. and Rabczuk, T. and Hackl, K.*Computer Methods in Applied Mechanics and Engineering*313 (2017)We in this paper present a novel adaptive finite element scheme for limit analysis of cracked structures. The key idea is to develop a general refinement algorithm based on a so-called polytree mesh structure. The method is well suited for arbitrary polygonal elements and furthermore traditional triangular and quadrilateral ones, which are considered as special cases. Also, polytree meshes are conforming and can be regarded as a generalization of quadtree meshes. For the aim of this paper, we restrict our main interest in plane-strain limit analysis to von Mises-type materials, yet its extension to a wide class of other solid mechanics problems and materials is completely possible. To avoid volumetric locking, we propose an approximate velocity field enriched with bubble functions using Wachspress coordinates on a primal-mesh and design carefully strain rates on a dual-mesh level. An adaptive mesh refinement process is guided by an L2-norm-based indicator of strain rates. Through numerical validations, we show that the present method reaches high accuracy with low computational cost. This allows us to perform large-scale limit analysis problems favorably. © 2016 Elsevier B.V.view abstract 10.1016/j.cma.2016.09.016 **A relaxation-based approach to damage modeling**

Junker, P. and Schwarz, S. and Makowski, J. and Hackl, K.*Continuum Mechanics and Thermodynamics*29 (2017)Material models, including softening effects due to, for example, damage and localizations, share the problem of ill-posed boundary value problems that yield mesh-dependent finite element results. It is thus necessary to apply regularization techniques that couple local behavior described, for example, by internal variables, at a spatial level. This can take account of the gradient of the internal variable to yield mesh-independent finite element results. In this paper, we present a new approach to damage modeling that does not use common field functions, inclusion of gradients or complex integration techniques: Appropriate modifications of the relaxed (condensed) energy hold the same advantage as other methods, but with much less numerical effort. We start with the theoretical derivation and then discuss the numerical treatment. Finally, we present finite element results that prove empirically how the new approach works. © 2016, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00161-016-0528-8 **A variational approach to the modelling of grooving in a three-dimensional setting**

Hackl, K. and Fischer, F.D. and Svoboda, J.*Acta Materialia*129 (2017)We present a theory of thermal grooving, i.e. surface motion due to surface diffusion, based solely on geometrical and energetic arguments and a variational approach involving a thermodynamic extremal principle. The theory is derived for a fully three-dimensional setting. All interface and contact conditions at junction lines and points of the material aggregate are derived rigorously and without ambiguity. A finite element implementation of the model is employed. Numerical examples are presented and compared with experimental results from the literature. © 2017 Acta Materialia Inc.view abstract 10.1016/j.actamat.2017.01.064 **Optimized growth and reorientation of anisotropic material based on evolution equations**

Jantos, D.R. and Junker, P. and Hackl, K.*Computational Mechanics*(2017)Modern high-performance materials have inherent anisotropic elastic properties. The local material orientation can thus be considered to be an additional design variable for the topology optimization of structures containing such materials. In our previous work, we introduced a variational growth approach to topology optimization for isotropic, linear-elastic materials. We solved the optimization problem purely by application of Hamilton’s principle. In this way, we were able to determine an evolution equation for the spatial distribution of density mass, which can be evaluated in an iterative process within a solitary finite element environment. We now add the local material orientation described by a set of three Euler angles as additional design variables into the three-dimensional model. This leads to three additional evolution equations that can be separately evaluated for each (material) point. Thus, no additional field unknown within the finite element approach is needed, and the evolution of the spatial distribution of density mass and the evolution of the Euler angles can be evaluated simultaneously. © 2017 Springer-Verlag GmbH Germanyview abstract 10.1007/s00466-017-1483-3 **A coupled dissipation functional for modeling the functional fatigue in polycrystalline shape memory alloys**

Waimann, J. and Junker, P. and Hackl, K.*European Journal of Mechanics, A/Solids*55 (2016)The pseudoelastic material behavior is one outstanding feature of shape memory alloys. This effect comes along with the forming of two plateaus in the stress/strain diagram of a tension test. Cyclic loading leads to a decrease particularly of the upper stress-plateau due to the evolution of plastic deformations which also implies fatigue of the material. In this work, we develop a variational material model which is able to predict the effect of fatigue using a novel approach for the dissipation potential that couples the evolutions of phase transformations and plastic strains. © 2015 Published by Elsevier Masson SAS.view abstract 10.1016/j.euromechsol.2015.08.001 **A discontinuous phase field approach to variational growth-based topology optimization**

Junker, P. and Hackl, K.*Structural and Multidisciplinary Optimization*54 (2016)Numerical instabilities cause the well-known problem of checkerboarding during topology optimization: elements that possess material are periodically neighbored to elements that are material-free. Furthermore, such numerical solutions depend on the finite element mesh and no reasonable processing techniques exist for manufacture. Thus, integral- or gradient-based regularization techniques are usually applied during topology optimization. In this paper, a novel approach to regularization is derived for a recently published variational approach to topology optimization that is based on material growth. The presented approach shares some similarities with the discontinuous Galerkin method and completely removes consideration of additional nodal quantities or complex integration schemes. The derivation and numerical treatment of the resulting phase field equation as well as exemplary numerical results are presented. © 2016, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00158-016-1398-1 **An evolutionary topology optimization approach with variationally controlled growth**

Jantos, D.R. and Junker, P. and Hackl, K.*Computer Methods in Applied Mechanics and Engineering*310 (2016)Previous works of Junker and Hackl (2016) have presented a variational growth approach to topology optimization in which the problem of checkerboarding was suppressed by means of a discontinuous regularization scheme. This approach did not require additional filter techniques and also optimization algorithms were not needed any more. However, growth approaches to topology optimization demand some limitations in order to avoid a global and simultaneous generation of mass. The limitation has been achieved by a rather simple approach with restricted possibilities for controlling. In this contribution, we eliminate this drawback by introducing a Lagrange multiplier to control the total mass within the model space for each iteration step. This enables us to achieve directly controlled growth behavior and even find optimized structures for prescribed structure volumes. Furthermore, a modified growth approach, which we refer to as the Lagrange shift approach, results a numerically stable model that is easy to handle. After the derivation of the approach, we present numerical solutions for different boundary problems that demonstrate the potential of our model. © 2016 Elsevier B.V.view abstract 10.1016/j.cma.2016.07.022 **Improved thermodynamic treatment of vacancy-mediated diffusion and creep**

Fischer, F.D. and Hackl, K. and Svoboda, J.*Acta Materialia*108 (2016)Approximately a decade ago a new concept to describe the kinetics of one-phase solid state systems evolving by diffusion and activity of vacancies has been published by the authors. The concept is based on the Onsager-Ziegler Thermodynamic Extremal Principle (TEP). In course of the last decade several improvements and corrections have been performed, which justify an overworking of the concept. A short introduction of the TEP is followed by a detail investigation of the Gibbs energy and its rate as well as of dissipation and dissipation function due to multicomponent diffusion process coupled with vacancy activity provoking swelling/shrinkage and creep and thus internal stress state development. The application of TEP allows an exact derivation of driving forces for the coupled processes. The Manning theory of diffusion is applied and the derivation of evolution equations for all system parameters (site fractions, swelling/shrinkage and creep strain tensor) is provided. © 2016 Acta Materialia Inc.view abstract 10.1016/j.actamat.2016.01.017 **Linking mathematics with engineering applications at an early stage – implementation, experimental set-up and evaluation of a pilot project**

Rooch, A. and Junker, P. and Härterich, J. and Hackl, K.*European Journal of Engineering Education*41 (2016)Too difficult, too abstract, too theoretical – many first-year engineering students complain about their mathematics courses. The project MathePraxis aims to resolve this disaffection. It links mathematical methods as they are taught in the first semesters with practical problems from engineering applications – and thereby shall give first-year engineering students a vivid and convincing impression of where they will need mathematics in their later working life. But since real applications usually require more than basic mathematics and first-year engineering students typically are not experienced with construction, mensuration and the use of engineering software, such an approach is hard to realise. In this article, we show that it is possible. We report on the implementation of MathePraxis at Ruhr-Universität Bochum. We describe the set-up and the implementation of a course on designing a mass damper which combines basic mathematical techniques with an impressive experiment. In an accompanying evaluation, we have examined the students' motivation relating to mathematics. This opens up new perspectives how to address the need for a more practically oriented mathematical education in engineering sciences. © 2015 SEFI.view abstract 10.1080/03043797.2015.1056095 **A mechanical model for dissolution-precipitation creep based on the minimum principle of the dissipation potential**

Klinge, S. and Hackl, K. and Renner, J.*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*471 (2015)In contrast to previous approaches that consider dissolution-precipitation creep as a multi-stage process and only simulate its governing subprocess, the present model treats this phenomenon as a single continuous process. The applied strategy uses the minimum principle of the dissipation potential according to which a Lagrangian consisting of elastic power and dissipation is minimized. Here, the elastic part has a standard form while the assumption for dissipation stipulates the driving forces to be proportional to two kinds of velocities: The material-transport velocity and the boundary-motion velocity. A Lagrange term is included to impose mass conservation. Two ways of solution are proposed. The strong form of the problem is solved analytically for a simple case. The weak form of the problem is used for a finite-element implementation and for simulating more complex cases. © 2015 The Author(s) Published by the Royal Society. All rights reserved.view abstract 10.1098/rspa.2014.0994 **A variational growth approach to topology optimization**

Junker, P. and Hackl, K.*Structural and Multidisciplinary Optimization*52 (2015)This paper presents a new approach to topology optimization that is based on observations of natural biological systems in which growth processes are initialized during high mechanical loading. A compliance parameter is introduced that serves as an internal variable and for which evolution equations are derived using the variational principle of the minimum of the dissipation potential. The well-known problem of checkerboarding is faced with regularization techniques on the Helmholtz free energy. The final procedure uses only the Helmholtz free energy as input. Several numerical examples are given for demonstration purposes. © 2015, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00158-015-1241-0 **A variational viscosity-limit approach to the evolution of microstructures in finite crystal plasticity**

Günther, C. and Junker, P. and Hackl, K.*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*471 (2015)A micromechanical model for finite single crystal plasticity was introduced by Kochmann & Hackl (2011 Contin.Mech. Thermodyn. 23, 63-85 (doi:10.1007/ s00161-010-0714-5)). This model is based on thermodynamic variational principles and leads to a non-convex variational problem. Based on the Lagrange functional, an incremental strategy was outlined to model the time-continuous evolution of a first-order laminate microstructure. Although this model provides interesting results on the material point level, owing to the global minimization in the evolution equations, the calculation time and numerical instabilities may cause problems when applying this model to macroscopic specimens. In this paper, a smooth transition zone between the laminates is introduced to avoid global minimization, which makes the numerical calculations cumbersome compared with the model in Kochmann & Hackl. By introducing a smooth viscous transition zone, the dissipation potential and its numerical treatment have to be adapted. We outline rate-dependent timeevolution equations for the internal variables based on variational techniques and show as first examples single-slip shear and tension/compression tests. © 2015 The Author(s) Published by the Royal Society. All rights reserved.view abstract 10.1098/rspa.2015.0110 **Rate-independent versus viscous evolution of laminate microstructures in finite crystal plasticity**

Günther, C. and Kochmann, D.M. and Hackl, K.*Lecture Notes in Applied and Computational Mechanics*78 (2015)In this chapter we investigate the variationalmodeling of the evolution of inelastic microstructures by the example of finite crystal plasticity with one active slip system. For this purpose we describe the microstructures by laminates of first order.We propose an analytical partial relaxation of an incompressible neo-Hookean energy formulation, keeping the internal variables and geometric microstructure parameters fixed, thus approximating the relaxed energy by an upper bound of the rank-one-convex hull. Based on the minimization of a Lagrange functional, consisting of the sum of rate of energy and dissipation potential, we derive an incremental strategy to model the time-continuous evolution of the laminate microstructure. Special attention is given to the three distinct cases of microstructure evolution, initiation, rotation, and continuous change. We compare a rate-independent approach with another one that employs viscous regularization which has certain advantages concerning the numerical implementation. Simple shear and tension/compression tests will be shown to demonstrate the differences between both approaches and to show the physical implications of the models introduced. © Springer International Publishing Switzerland 2015.view abstract 10.1007/978-3-319-18242-1_3 **Variational prediction of the mechanical behavior of shape memory alloys based on thermal experiments**

Junker, P. and Jaeger, S. and Kastner, O. and Eggeler, G. and Hackl, K.*Journal of the Mechanics and Physics of Solids*80 (2015)In this work, we present simulations of shape memory alloys which serve as first examples demonstrating the predicting character of energy-based material models. We begin with a theoretical approach for the derivation of the caloric parts of the Helmholtz free energy. Afterwards, experimental results for DSC measurements are presented. Then, we recall a micromechanical model based on the principle of the minimum of the dissipation potential for the simulation of polycrystalline shape memory alloys. The previously determined caloric parts of the Helmholtz free energy close the set of model parameters without the need of parameter fitting. All quantities are derived directly from experiments. Finally, we compare finite element results for tension tests to experimental data and show that the model identified by thermal measurements can predict mechanically induced phase transformations and thus rationalize global material behavior without any further assumptions. © 2015 Elsevier Ltd.view abstract 10.1016/j.jmps.2015.04.015 **A model for high temperature creep of single crystal superalloys based on nonlocal damage and viscoplastic material behavior**

Trinh, B.T. and Hackl, K.*Continuum Mechanics and Thermodynamics*26 (2014)A model for high temperature creep of single crystal superalloys is developed, which includes constitutive laws for nonlocal damage and viscoplasticity. It is based on a variational formulation, employing potentials for free energy, and dissipation originating from plasticity and damage. Evolution equations for plastic strain and damage variables are derived from the well-established minimum principle for the dissipation potential. The model is capable of describing the different stages of creep in a unified way. Plastic deformation in superalloys incorporates the evolution of dislocation densities of the different phases present. It results in a time dependence of the creep rate in primary and secondary creep. Tertiary creep is taken into account by introducing local and nonlocal damage. Herein, the nonlocal one is included in order to model strain localization as well as to remove mesh dependence of finite element calculations. Numerical results and comparisons with experimental data of the single crystal superalloy LEK94 are shown. © 2013 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00161-013-0317-6 **A thermo-mechanically coupled field model for shape memory alloys**

Junker, P. and Hackl, K.*Continuum Mechanics and Thermodynamics*26 (2014)The impressive properties of shape memory alloys are produced by means of solid-to-solid phase transformations where thermal effects play an important role. In this paper, we present a model for polycrystalline shape memory alloys which takes full thermo-mechanical coupling into account. Starting from the equations of the first and the second law of thermodynamics, we derive evolution equations for the volume fractions of the different martensitic variants and a related equation for heat conduction. A thermodynamic analysis allows to formulate a complete expression for the dissipation caused by phase transformation and heat flux. This allows to model the experimentally well-documented transformation fronts in tension tests by a finite element scheme without further assumptions. Additionally, the number of required model parameters is very small in comparison with phenomenological approaches. Numerical examples are presented which show a good agreement with experimental observations. © 2014, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00161-014-0345-x **Coupled computational simulation of excavation and soil transport in earth-pressure balance shield tunneling machines using a viscous two-phase fluid model for soil-foam mixtures**

Dang, T.S. and Meschke, G. and Wessels, N. and Hackl, K.*Geotechnical Special Publication*(2014)The excavation process ofearth-pressure balance (EPB) shield machines involves the cutting of the ground at the tunnel face and the transport of the soil paste in the excavation chamber. For the numerical simulation of these two processes, a computational strategy, characterized by the coupling of two partial models using the discrete element method (DEM) and the finite element method (FEM) has been developed (Wessels et al., 2013). Excavation is simulated using DEM, with the fracture process being represented by the release of interaction forces between the particles. The transport of the excavated soil mixed with the soil conditioning foam, yielding a pasty soil-foam mixture within the pressure chamber, is simulated by means of a two-phase fluid model in Eulerian description. The two momentum and mixture mass equations are discretized in time by the characteristic-based split method (CBS) (Zienkiewicz et al., 2005). The phase volume fraction equations are solved using upwind weighting functions according to the improved Mizukami-Hughes method (Knobloch, 2006). The proposed model is applied to a coupled analysis of the excavation and transport-mixing flow inside a simplified pressure chamber with foam injections. The mixing process, due to the rotation in the chamber, is preliminarily investigated by the mixing flow in a 2D cavity test case. © 2014 American Society of Civil Engineers.view abstract 10.1061/9780784413449.076 **The principle of the minimum of the dissipation potential for non-isothermal processes**

Junker, P. and Makowski, J. and Hackl, K.*Continuum Mechanics and Thermodynamics*26 (2014)In this paper, we contribute to the methodology of material modeling by presenting a potential-based approach for non-isothermal inelastic processes. It is based on the principle of the minimum of the dissipation potential which was used previously only in the isothermal context. In contrast to the principle of maximum dissipation, the presented procedure results in mathematically simplified equations. Due to its variational character, the inclusion of constraints is very simple. After derivation of our method, we use the examples of non-isothermal perfect plasticity and shape memory alloys for demonstration of the validity and performance of the concept. © 2013 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00161-013-0299-4 **A variational approach to grooving and wetting**

Hackl, K. and Fischer, F.D. and Klevakina, K. and Renner, J. and Svoboda, J.*Acta Materialia*61 (2013)Two bodies, e.g. grains with a certain surface contour, are assumed to be in contact at a plane interface, e.g. a common grain boundary with an arbitrary inclination relatively to the surface and with zero mobility and diffusivity. A groove appears due to surface diffusion along the triple line, i.e. the intersection line of the two surfaces and the grain boundary. The thermodynamic extremum principle is applied to derive the evolution equations for the surfaces of both bodies as well as the contact conditions at the triple line. Applications to grooving and wetting are demonstrated and compared with the results from the literature. The simulations indicate that the groove root angle can be significantly different from the value of the dihedral angle calculated from the equilibrium condition for the specific grain boundary and surface energies. Moreover, it is demonstrated that the groove angle is dependent on the kinetic parameters, e.g. surface diffusion coefficients of individual grains. © 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.view abstract 10.1016/j.actamat.2012.11.035 **High-temperature deformation and recrystallization: A variational analysis and its application to olivine aggregates**

Hackl, K. and Renner, J.*Journal of Geophysical Research: Solid Earth*118 (2013)We develop a framework for a variational analysis of microstructural evolution during inelastic high-temperature deformation accommodated by dislocation mechanisms and diffusive mass transport. A polycrystalline aggregate is represented by a distribution function characterizing the state of individual grains by three variables, dislocation density, grain size, and elastic strain. The aggregate's free energy comprises elastic energy and energies of lattice distortions due to dislocations and grain boundaries. The work performed by the external loading is consumed by changes in the number of defects and their migration leading to inelastic deformation. The variational approach minimizes the rate of change of free energy with the evolution of the state variables under constraints on the aggregate volume, on a relation between changes in plastic strain and dislocation density, and on the form of the dissipation functionals for defect processes. The constrained minimization results in four basic evolution equations, one each for the evolution in grain size and dislocation density and flow laws for dislocation and diffusion creep. Analytical steady state scaling relations between stress and dislocation density and grain size (piezometers) are derived for quasi-homogeneous materials characterized by a unique relation between grain size and dislocation density. Our model matches all currently available experimental observations regarding high-temperature deformation of olivine aggregates with plausible values for the involved micromechanical model parameters. The relation between strain rate and stress for olivine aggregates maintaining a steady state microstructure is distinctly nonlinear in stark contrast to the majority of geodynamical modeling relying on linear relations, i.e., Newtonian behavior. Key Points Analytical derivation of steady-state piezometers using variational analysis Matches observations for olivine rocks with plausible micromechanical parameters Provides insight into rheology of olivine aggregates, e.g., lifetime of grains ©2013. American Geophysical Union. All Rights Reserved.view abstract 10.1002/jgrb.50125 **Investigation of the influence of reflection on the attenuation of cancellous bone**

Klinge, S. and Hackl, K. and Gilbert, R.P.*Biomechanics and Modeling in Mechanobiology*12 (2013)The model proposed in this paper is based on the fact that the reflection might have a significant contribution to the attenuation of the acoustic waves propagating through the cancellous bone. The numerical implementation of the mentioned effect is realized by the development of a new representative volume element that includes an infinitesimally thin 'transient' layer on the contact surface of the bone and the marrow. This layer serves to model the amplitude transformation of the incident wave by the transition through media with different acoustic impedances and to take into account the energy loss due to the reflection. The proposed representative volume element together with the multiscale finite element is used to simulate the wave propagation and to evaluate the attenuation coefficient for samples with different effective densities in the dependence of the applied excitation frequency. The obtained numerical values show a very good agreement with the experimental results. Moreover, the model enables the determination of the upper and the lower bound for the attenuation coefficient. © 2012 Springer-Verlag.view abstract 10.1007/s10237-012-0391-x **A model for the evolution of laminates in finite-strain elastoplasticity**

Hackl, K. and Heinz, S. and Mielke, A.*ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik*92 (2012)We study the time evolution in elastoplasticity within the rate-independent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of time-incremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented. The authors study the time evolution in elastoplasticity within the rate-independent framework of generalized standard materials. Their particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, they will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, they hall allow for such microstructure that is given by simple or sequential laminates. The authors will consider a model for the evolution of these laminates and will prove a theorem on the existence of solutions to any finite sequence of time-incremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented. Copyright © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/zamm.201100155 **Application of the multiscale fem to the modeling of nonlinear composites with a random microstructure**

Klinge, S. and Hackl, K.*International Journal for Multiscale Computational Engineering*10 (2012)In this contribution the properties and application of the multiscale finite element program MSFEAP are presented. This code is developed on basis of coupling the homogenization theory with the finite element method. According to this concept, the investigation of an appropriately chosen representative volume element yields the material parameters needed for the simulation of a macroscopic body. The connection of scales is based on the principle of volume averaging and the Hill-Mandel macrohomogeneity condition. The latter leads to the determination of different types of boundary conditions for the representative volume element and in this way to the postulation of a well-posed problem at this level. The numerical examples presented in the contribution investigate the effective material behavior of microporous media. An isotropic and a transversally anisotropic microstructure are simulated by choosing an appropriate orientation and geometry of the representative volume element in each Gauss point. The results are verified by comparing them with Hashin-Shtrikman's analytic bounds. However, the chosen examples should be understood as simply an illustration of the program application, while its main feature is a modular structure suitable for further development. © 2012 by Begell House, Inc.view abstract 10.1615/IntJMultCompEng.2012002059 **Contribution of the reflection to the attenuation properties of cancellous bone**

Klinge, S. and Hackl, K.*Complex Variables and Elliptic Equations*57 (2012)The article deals with the contribution of reflection effects to the attenuation properties of cancellous bone. The bone behaviour is simulated by the multiscale finite element method, a numerical homogenization approach, suitable for the modelling of heterogeneous material with a highly oscillatory microstructure. The focus is on the modelling of a novel type of the representative volume element, which apart from the solid framework filled with fluid marrow also includes an infinitesimally thin 'transition' layer at the contact of the phases. The mentioned layer is implemented in order to simulate the amplitude transformation of an incident wave and to take the loss of energy caused by the reflection into account. The given numerical examples consider the simulation of wave propagation through a sample while the excitation frequency is varied. The numerical values are compared with the results which are determined without considering the reflection, in order to point out the contribution of the newly introduced phenomenon. © 2012 Copyright Taylor and Francis Group, LLC.view abstract 10.1080/17476933.2011.617011 **Generation and evolution of inelastic microstructures - An overview**

Hackl, K. and Hoppe, U. and Kochmann, D.M.*GAMM Mitteilungen*35 (2012)In this paper we give an overview on the modeling of inelastic microstructures using variational methods. We start by discussing the underlying variational principles for inelastic materials, derive evolution equations for internal variables, and introduce the concept of condensed energy. As a mathematical prerequisite we review the variational calculus of nonconvex potentials and the notion of relaxation. We use these instruments in order to study the initiation of plastic microstructures. Here we focus on a model of single-slip crystal plasticity. Afterwards we move on to model the evolution of microstructures. We introduce the concept of essential microstructures and the corresponding relaxed energies and dissipation potentials, and derive evolution equations for microstructure parameters. We then present a numerical scheme by means of which the microstructure development can be computed, and show numerical results for particular examples. ©c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201210007 **Modelling the kinetics of a triple junction**

Fischer, F.D. and Svoboda, J. and Hackl, K.*Acta Materialia*60 (2012)The grain structure in a one phase system involves grain boundaries (surfaces), three-grain junctions (lines) and four-grain junctions (points). A certain Gibbs energy and mobility can be assigned to each object. System evolution, driven by a decrease in the total Gibbs energy, occurs by migration of objects constrained by rather complex contact conditions. A system with cylindrical symmetry is assumed, where three grain boundaries with different mobilities and different specific Gibbs energies are in contact at a triple junction line of given mobility. The equations of evolution of the system are derived by means of the thermodynamic extremal principle. New general contact conditions at the triple junction are derived, including the mobilities of all objects and the energies, contact angles and curvatures of the grain boundaries. Special contact conditions are also provided for the cases where the triple junction mobility is infinite and/or in the case of one of the grain boundaries having zero mobility. The model is demonstrated by several examples. © 2012 Acta Materialia Inc. Published by Elsevier Ltd.view abstract 10.1016/j.actamat.2012.05.018 **Performance of mixed and enhanced finite elements for strain localization in hypoplasticity**

Trinh, B.T. and Hackl, K.*International Journal for Numerical and Analytical Methods in Geomechanics*36 (2012)Displacement and mixed finite element formulations of shear localization in materials are presented. The formulations are based on hypoplastic constitutive laws for soils and the mixed enhanced treatment involving displacement, strain and stress rates as independently varied fields. Included in these formulations are the standard displacement method, the three-field mixed formulation, the enhanced assumed strain method and the mixed enhanced strain method. Several numerical examples demonstrating the capability and performance of the different finite element formulations are presented. The numerical results are compared with available experimental data for Hostun RF sand and numerical results for Karlsruhe sand on biaxial tests. © 2011 John Wiley & Sons, Ltd.view abstract 10.1002/nag.1042 **The influence of particle size and spacing on the fragmentation of nanocomposite anodes for Li batteries**

Dimitrijevic, B.J. and Aifantis, K.E. and Hackl, K.*Journal of Power Sources*206 (2012)Experimental evidence has shown that composites comprised Si and Sn nanoparticles embedded inside a matrix are the most promising next generation anodes for Li-ion batteries. This is due to the ability of the matrix material to constrain/buffer the up to 300 volume expansion that Sn and Si undergo upon the formation of lithium rich alloys. Damage still occurs at the nanoparticle/matrix interface, and hence further materials design is required in order to commercialize such anodes. Initial theoretical works have predicted that low volume fractions and high aspect ratios of the nanoparticles result in a greater mechanical stability and hence better capacity retention. The most important design parameters, however, such as particle size and spacing have not been considered theoretically. In the present study, therefore, a gradient enhanced damage model will be employed to predict that damage during Li-insertion, is negligible when the particle size is 20 nm, and the interparticle half-spacing greater then 1.5 times the particle diameter. Furthermore, from the matrix materials considered herein graphene is predicted to be the most promising matrix, which is consistent with recent experimental data. © 2012 Elsevier B.V. All rights reserved.view abstract 10.1016/j.jpowsour.2012.01.065 **A regularization framework for damage-plasticity models via gradient enhancement of the free energy**

Dimitrijevic, B.J. and Hackl, K.*International Journal for Numerical Methods in Biomedical Engineering*27 (2011)We present a framework for the regularization of coupled damage-plasticity models through gradient enhancement of the Helmholtz-free-energy function. This enhancement is achieved introducing new variables, coupled to the gradients of the inelastic variables and thus regularizing the model. The variational formulation of the model results in a pure minimization problem. Numerical examples, which show the performance of the proposed gradient-enhanced model, are presented. It is shown that the pathological mesh dependence of the coupled inelastic model is efficiently removed, together with the difficulties of numerical calculations in the softening range. © 2009 John Wiley & Sons, Ltd.view abstract 10.1002/cnm.1350 **A study on the principle of maximum dissipation for coupled and non-coupled non-isothermal processes in materials**

Hackl, K. and Fischer, F.D. and Svoboda, J.*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*467 (2011)Onsager's principle of maximum dissipation (PMD) has proven to be an efficient tool to derive evolution equations for the internal variables describing non-equilibrium processes. However, a rigorous treatment of PMD for several simultaneously acting dissipative processes is still open and presented in this paper. The coupling or uncoupling of the processes is demonstrated via the mathematical structure of the dissipation function. Examples are worked out for plastic deformation and heat flux.view abstract 10.1098/rspa.2010.0179 **A study on the principle of maximum dissipation for coupled and non-coupled non-isothermal processes in materials (vol 467, pg 1186, 2010)**

Hackl, K. and Fischer, F. D. and Svoboda, J.*Proceedings of the Royal Society A-mathematical Physical and Engineering Sciences*467 (2011)view abstract 10.1098/rspa.2011.0015 **Application of a biphasic representative volume element to the simulation of wave propagation through cancellous bone**

Ilic, S. and Hackl, K. and Gilbert, R.P.*Journal of Computational Acoustics*19 (2011)This paper deals with the application of the multiscale finite element method for simulating the cancellous bone. For this purpose, two types of biphasic representative volume elements are proposed. In the first one, the solid frame consists of thin walls simulated by shell elements. On the other hand, the solid phase of the second model is made up of columns consisting of eight-node brick elements. This choice of representative volume elements is motivated by experimental investigations reporting on the existence of plate-like and rode-like types of cancellous bone and possible conversions between them. The proposed representative volume elements are used to calculate effective material tensors and parameters and to investigate their change in terms of increasing porosity, which is typical for osteoporosis. As a first example, changes in the geometry of the representative volume elements are used to explore material anisotropy. In the end, the final example considers wave propagation through the bone treated as a homogenized medium. © 2011 IMACS.view abstract 10.1142/S0218396X11004407 **Finite element simulations of poly-crystalline shape memory alloys based on a micromechanical model**

Junker, P. and Hackl, K.*Computational Mechanics*47 (2011)We present a finite element implementation of a micromechanically motivated model for poly-crystalline shape memory alloys, based on energy minimization principles. The implementation allows simulation of anisotropic material behavior as well as the pseudo-elastic and pseudoplastic material response of whole samples. The evolving phase distribution over the entire specimen is calculated. The finite element model predicts the material properties for a relatively small number of grains. For different points of interest in the specimen the model can be consistently evaluated with a significantly higher number of grains in a post-processing step, which allows to predict the re-orientation of martensite at different loads. The influence of pre-texture on the material's properties, due to some previous treatment like rolling, is discussed. © Springer-Verlag 2010.view abstract 10.1007/s00466-010-0555-4 **The evolution of laminates in finite crystal plasticity: A variational approach**

Kochmann, D.M. and Hackl, K.*Continuum Mechanics and Thermodynamics*23 (2011)The analysis and simulation of microstructures in solids has gained crucial importance, virtue of the influence of all microstructural characteristics on a material's macroscopic, mechanical behavior. In particular, the arrangement of dislocations and other lattice defects to particular structures and patterns on the microscale as well as the resultant inhomogeneous distribution of localized strain results in a highly altered stress-strain response. Energetic models predicting the mechanical properties are commonly based on thermodynamic variational principles. Modeling the material response in finite strain crystal plasticity very often results in a non-convex variational problem so that the minimizing deformation fields are no longer continuous but exhibit small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. This results in fine structures that can be interpreted as the observed microstructures. In this paper, we first review the underlying variational principles for inelastic materials. We then propose an analytical partial relaxation of a Neo-Hookean energy formulation, based on the assumption of a first-order laminate microstructure, thus approximating the relaxed energy by an upper bound of the rank-one-convex hull. The semi-relaxed energy can be employed to investigate elasto-plastic models with a single as well as multiple active slip systems. Based on the minimization of a Lagrange functional (consisting of the sum of energy rate and dissipation potential), we outline an incremental strategy to model the time-continuous evolution of the laminate microstructure, then present a numerical scheme by means of which the microstructure development can be computed, and show numerical results for particular examples in single- and double-slip plasticity. We discuss the influence of hardening and of slip system orientations in the present model. In contrast to many approaches before, we do not minimize a condensed energy functional. Instead, we incrementally solve the evolution equations at each time step and account for the actual microstructural changes during each time step. Results indicate a reduction in energy when compared to those theories based on a condensed energy functional. © 2010 Springer-Verlag.view abstract 10.1007/s00161-010-0174-5 **Variational modeling of shape memory alloys - An overview**

Hackl, K. and Junker, P. and Heinen, R.*International Journal of Materials Research*102 (2011)Shape memory alloys can be described in a uniform way relying on energetic considerations only. We present micromechanically motivated models for single and polycrystals. The approach studied here is based on energy minimization and includes hysteretic effects via a simple dissipation ansatz. It is capable of reproducing important aspects of the material behavior such as pseudoelasticity and pseudoplasticity. The influence of anisotropies in the crystalline texture as well as in the elastic constants of the austenite and the martensitic variants is also discussed. Furthermore, regularization is applied in order to receive localized but still mesh independent results for phase distributions in a finite element implementation. The entire presentation emphasizes the usage of variational methods leading to the notion of relaxed potentials. Interrelations to various other applications of these concepts will be highlighted. © Carl Hanser Verlag GmbH & Co. KG.view abstract 10.3139/146.110527 **A micromechanical model for polycrystalline shape memory alloys - Formulation and numerical validation**

Heinen, R. and Hackl, K.*IUTAM Bookseries*21 (2010)The specific material properties of shape memory alloys are due to the formation of martensitic microstructures. In this contribution, we develop a strategy to model the material behavior based on energy considerations: we first present narrow bounds to the elastic energy obtained by lamination of the multi-well problem in the monocrystalline case. These considerations are then extended to polycrystals and compared to a convexification bound. Due to the acceptably low difference between convexification lower and lamination upper bound,we use the convexification bound to establish a micromechanical model which, on the basis of physically well motivated parameters such as elastic constants and transformation strains, is able to represent a variety of aspects of the material behavior such as seudoelasticity, pseudoplasticity and martensite reorientation. © 2010 Springer Science+Business Media B.V.view abstract 10.1007/978-90-481-9195-6-7 **Application of the multiscale FEM to the modeling of cancellous bone**

Ilic, S. and Hackl, K. and Gilbert, R.*Biomechanics and Modeling in Mechanobiology*9 (2010)This paper considers the application of multiscale finite element method (FEM) to the modeling of cancellous bone as an alternative for Biot's model, the main intention of which is to decrease the extent of the necessary laboratory tests. At the beginning, the paper gives a brief explanation of the multiscale concept and thereafter focuses on the modeling of the representative volume element and on the calculation of the effective material parameters, including an analysis of their change with respect to increasing porosity. The latter part of the paper concentrates on the macroscopic calculations, which is illustrated by the simulation of ultrasonic testing and a study of the attenuation dependency on material parameters and excitation frequency. The results endorse conclusions drawn from the experiments: increasing excitation frequency and material density cause increasing attenuation. © 2009 Springer-Verlag.view abstract 10.1007/s10237-009-0161-6 **Configurational forces and couples in fracture mechanics accounting for microstructures and dissipation**

Stumpf, H. and Makowski, J. and Hackl, K.*International Journal of Solids and Structures*47 (2010)Configurational forces and couples acting on a dynamically evolving fracture process region as well as their balance are studied with special emphasis to microstructure and dissipation. To be able to investigate fracture process regions preceding cracks of mode I, II and III we choose as underlying continuum model the polar and micropolar, respectively, continuum with dislocation motion on the microlevel. As point of departure balance of macroforces, balance of couples and balance of microforces acting on dislocations are postulated. Taking into account results of the second law of thermodynamics the stress power principle for dissipative processes is derived. Applying this principle to a fracture process region evolving dynamically in the reference configuration with variable rotational and crystallographic structure, the configurational forces and couples are derived generalizing the well-known Eshelby tensor. It is shown that the balance law of configurational forces and couples reflects the structure of the postulated balance laws on macro- and microlevel: the balance law of configurational forces and configurational couples are coupled by field variable, while the balance laws of configurational macro- and microforces are coupled only by the form of the free energy. They can be decoupled by corresponding constitutive assumption. Finally, it is shown that the second law of thermodynamics leads to the result that the generalized Eshelby tensor for micropolar continua with dislocation motion consists of a non-dissipative part, derivable from free and kinetic energy, and a dissipative part, derivable from a dissipation pseudo-potential. © 2010 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijsolstr.2010.04.032 **Integrity of micro-hotplates during high-temperature operation monitored by digital holographic microscopy**

Lai, Y.W. and Koukourakis, N. and Gerhardt, N.C. and Hofmann, M.R. and Meyer, R. and Hamann, S. and Ehmann, M. and Hackl, K. and Darakis, E. and Ludwig, Al.*Journal of Microelectromechanical Systems*19 (2010)An investigation on the integrity of micro-hotplates using in situ digital holographic microscopy is reported. The surface topography and surface evolution of the devices during high-temperature operation (heating/cooling cycles) is measured with nanometer-scale resolution. A localized permanent out-of-plane surface deformation of 40% of the membrane thickness caused by the top measurement electrodes occurring after the first cycle is observed. The integrity-related issues caused by such a permanent deformation are discussed. © 2006 IEEE.view abstract 10.1109/JMEMS.2010.2067442 **Solution-Precipitation Creep - Modeling and Extended FE Implementation**

Ilic, S. and Hackl, K.*Iutam Symposium on Variational Concepts with Applications to the Mechanics of Materials*21 (2010)The topic of this contribution is the mechanical modeling of solution-precipitation creep, a process occurring in polycrystalline and granular structures under specific temperature and pressure conditions. The model presented has a variational structure and is based on a novel proposal for the dissipation while the elastic energy is kept in the standard form. The assumed dissipation term depends on two kinds of velocities characteristic for the process: velocity of material transfer and velocity of inelastic deformations, both manifesting themselves on the boundaries of the grains. For the numerical implementation, the standard finite element program FEAP together with the pre- and postprocessing software package GID are used. The simulations are illustrated by two examples, a polycrystal with regular hexagonal microstructure and a polycrystal with random microstructure.view abstract 10.1007/978-90-481-9195-6_8 **Time-Continuous Evolution of Microstructures in Finite Plasticity**

Kochmann, D. M. and Hackl, K.*Iutam Symposium on Variational Concepts with Applications to the Mechanics of Materials*21 (2010)Plastic deformation of crystalline solids very often gives rise to the initation of material microstructures experimentally visible as dislocation patterns. These microstructures are not inherent to the material but occur as a result of deformation. Modeling a physically deformed crystal in finite plasticity by means of the displacement field and in terms of a set of internal variables which capture the microstructural characteristics, we employ energy principles to analyze the microstructure formation and evolution as a result of energy minimization. In particular, for non-quasiconvex energy potentials the minimizers are no longer continuous deformation fields but small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. We briefly review the variational concept of the underlying energy principles for inelastic materials. As a first approximation of the relaxed energy density, we assume first-order laminate microstructures, thus approximating the relaxed energy by the rank-one convex envelope. Based on this approach, we present explicit time-evolution equations for the volume fractions and the internal variables, then outline a numerical scheme by means of which the microstructure evolution can be computed and we show numerical results for particular examples in single and double-slip plasticity. In contrast to many approaches before we do not globally minimize a condensed energy functional to determine the microstructure but instead incrementally solve the evolution equations at each time step, in particular accounting for the dissipation required to rearrange the microstructure during a finite time increment with already existing mictrostructure at the beginning of the time step.view abstract 10.1007/978-90-481-9195-6_9 **Variational concepts with applications to microstructural evolution**

Fischer, F.D. and Svoboda, J. and Hackl, K.*IUTAM Bookseries*21 (2010)In systems at elevated temperature the development of the microstructure of a material is controlled by diffusional and interface migration processes. As first step the description of the microstructure is reduced to a finite number of time-dependent characteristic parameters (CPs). Then the Thermodynamic Extremal Principle (TEP) is engaged to develop the evolution equations for these characteristic parameters. This treatment is demonstrated on a bamboo-structured material system predicting the spatial and time distribution of chemical composition as well as the deformation state. © 2010 Springer Science+Business Media B.V.view abstract 10.1007/978-90-481-9195-6-6

#### composites

#### finite element method

#### mechanical properties

#### microstructure

#### numerical methods

#### structural mechanics