#### Prof. Dr.-Ing. Jörn Mosler

Institute of Mechanics

TU Dortmund University

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- joern[dot]mosler[at]tu-dortmund[dot]de
- +49 231 755 5744/7978
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**How regularization concepts interfere with (quasi-)brittle damage: a comparison based on a unified variational framework**

Langenfeld, K. and Kurzeja, P. and Mosler, J.*Continuum Mechanics and Thermodynamics*34 (2022)Three regularization concepts are assessed regarding their variational structure and interference with the predicted physics of (quasi-)brittle damage: the fracture energy concept, viscous regularization and micromorphic regularization. They are first introduced in a unified variational framework, depicting how they distinctively evolve from incremental energy minimization. The analysis of a certain time interval of a one-dimensional example is used to show how viscous and micromorphic regularization retains well-posedness within the softening regime. By way of contrast, the fracture energy concept is characterized by ill-posedness—as known from previous non-variational analyses. Numerical examples finally demonstrate the limitations and capabilities of each concept. The ill-posed local fracture energy concept leads by its design to a spatially constant fracture energy—in line with Griffith’s theory. The viscous regularization, in turn, yields a well-posed problem but artificial viscosity can add a bias to unloading and fracture thickness. Furthermore, and even more important, a viscous regularization does not predict a spatially constant fracture energy due to locally heterogeneous loading rates. The well-posed micromorphic regularization is in line with the underlying physics and does not show this undesired dependency. However, it requires the largest numerical efforts, since it is based on a coupled two-field formulation. © 2022, The Author(s).view abstract 10.1007/s00161-022-01143-2 **Cahn–Hilliard phase field theory coupled to mechanics: Fundamentals, numerical implementation and application to topology optimization**

Bartels, A. and Kurzeja, P. and Mosler, J.*Computer Methods in Applied Mechanics and Engineering*383 (2021)The presented framework couples the Cahn–Hilliard phase field theory to continuum mechanics using a variational principle. All equations follow consistently from stationary of a rate potential and yield a physically sound homogenization. Static and kinematic compatibility at the material interfaces are naturally guaranteed. In order to enforce admissibility of the phase field parameter, nonlinear complementary conditions are considered and embedded into the algorithmic formulation. Eventually, the variationally consistent framework also features topology optimization automatically. In contrast to other approaches that start from the optimization problem, the present formulation starts from a more comprehensive energy potential. This perspective allows to explore the natural physical mechanisms that control the system's compliance (e.g., interface evolution) and that drive maximum structural performance (changing the direction of the evolution equation with respect to the phase field parameter). Furthermore, this perspective efficiently couples the physical constraints (e.g., mass and momentum conservation). Energetically optimized microstructures and an optimized beam structure illustrate the applicability as well as the numerical performance of the elaborated framework. © 2021 Elsevier B.V.view abstract 10.1016/j.cma.2021.113918 **On general imperfect interfaces with spatially non-constant displacement jumps**

Heitbreder, T. and Kurzeja, P. and Mosler, J.*International Journal of Solids and Structures*232 (2021)A novel constitutive framework suitable for material interfaces undergoing large deformations is presented. In contrast to previous works, it accounts for spatially non-constant displacement jumps in the Helmholtz energy by employing their gradient along the interface. This first-order generalization encompasses classic cohesive zone models and surface elasticity theory as special cases. Based on a unifying variational ansatz, all balance laws are derived in a natural manner. Balance of angular momentum is enforced pointwise by designing energies that are material frame indifferent. It is shown that for quadratic energies already existing interface models can predict the same features as the novel framework. However, for higher-order energies, this analogy is lost and the generalized, gradient-based framework indeed allows to capture additional effects. An exemplary experiment highlights such effects for a 3d-printed specimen with a soft interface. © 2021 Elsevier Ltdview abstract 10.1016/j.ijsolstr.2021.111068 **Projection vs. relaxation of adjacent bulk deformation for surface modeling: Theoretical and numerical aspects**

Sievers, C. and Mosler, J. and Kurzeja, P.*International Journal of Solids and Structures*226-227 (2021)Motivated by a strong coupling between bulk and surface physics, we present two approaches to derive continuum surface models from the three-dimensional, adjacent bulk deformation: projection and relaxation. In contrast to conventional ad hoc models, properties like the surface stress are consistently derived from a hyperelastic thermodynamic potential. While the projection approach captures classical in-plane stresses, the relaxation approach can further relax normal-normal and normal-shear coupling. A projection onto the surface is indeed always a relaxation if anisotropy is superficial, but not vice versa. The distinct behavior of projection and relaxation is theoretically discussed and highlighted by specific examples. Numerical implementation in a finite-element framework is subsequently elaborated. Its performance is illustrated by isotropic and anisotropic surface energies of a free cube and a beam under tension. Finally, both projection and relaxation constitute valuable alternatives to conventional surface models in terms of physics, geometry and computation. © 2021 Elsevier Ltdview abstract 10.1016/j.ijsolstr.2021.111084 **A micromorphic approach for gradient-enhanced anisotropic ductile damage**

Langenfeld, K. and Mosler, J.*Computer Methods in Applied Mechanics and Engineering*360 (2020)This paper deals with the numerically effective modeling of anisotropic material degradation caused by ductile damage. Although standard local anisotropic damage models are relatively well-developed nowadays, their regularization which is required in order to eliminate their mathematical ill-posedness is far from being straightforward. It bears emphasis that this regularization is not only required from a mathematical point of view, since the aforementioned ill-posedness is known to be the source for the pathological mesh dependence as far as the finite element method is concerned. Within this paper, a general local framework for capturing anisotropic material degradation caused by ductile damage is extended to a non-local model by means of a gradient-enhancement. However, in order to achieve a numerically effective implementation, the gradient-enhancement is not implemented in a direct manner, but by means of a micromorphic approximation. By doing so, the implementation of the underlying local model is almost unaffected. Particularly the inequalities resulting from the yield function can be restricted to the local integration point level. Since a naive micromorphic implementation turns out to be unsuitable for regularizing the underlying local model, a novel adaption of the yield function is proposed. It is shown that the resulting single-surface-model is indeed able to capture anisotropic material degradation caused by ductile damage and, furthermore, that the finite element implementation is mesh objective. © 2019 Elsevier B.V.view abstract 10.1016/j.cma.2019.112717 **Computational homogenization of material surfaces: From atomistic simulations to continuum models**

Sievers, C. and Mosler, J. and Brendel, L. and Kurzeja, P.*Computational Materials Science*175 (2020)The objective of this work is a numerical multiscale framework that determines mechanical continuum properties of material surfaces based on molecular statics. The key idea is the coupling of representative volume elements in the atomistic and in the continuum model by the principle of energy equivalence. This allows a thermodynamically consistent implementation of various material models and boundary conditions, e.g., to capture size effects in nano scale materials. For the present example of copper, we observe a very good match with literature data. Only the results for the surface stiffness still deviate in the same range as existing data sources do. The presented results concurrently indicate a drastic strain sensitivity. We further eliminate a methodological bulk to surface error propagation by an appropriate strain limit and thickness extrapolation. The latter is calculated by always allowing for fully developed surface regions. Additionally, our method reveals a strain dependence of higher order that is caused by the anharmonic potential and not captured by standard bulk models. The presented multiscale framework finally serves two purposes: validating the reasonableness of a material surface model and determining its parameters. © 2019 Elsevier B.V.view abstract 10.1016/j.commatsci.2019.109431 **Influence of anisotropic damage evolution on cold forging**

Langenfeld, K. and Schowtjak, A. and Schulte, R. and Hering, O. and Möhring, K. and Clausmeyer, T. and Ostwald, R. and Walther, F. and Tekkaya, A.E. and Mosler, J.*Production Engineering*14 (2020)This contribution deals with the influence of anisotropic material degradation (damage) within numerical simulations of cold forging. For that purpose, two constitutive frameworks for modeling ductile damage are presented: an isotropic and an anisotropic model. In a first step, both models are calibrated based on a uniaxial tensile test. Then, the forward rod extrusion process is simulated with the isotropic model. The deformation of a characteristic element is transferred to the anisotropic model and the local response is investigated. Both models are compared to one another in terms of the process induced ductile damage. It will be shown, that the magnitude of the induced damage agrees reasonably well, but that the orientation of ductile damage is of major importance. © 2020, German Academic Society for Production Engineering (WGP).view abstract 10.1007/s11740-019-00942-y **Classic crystal plasticity theory vs crystal plasticity theory based on strong discontinuities—Theoretical and algorithmic aspects**

Fohrmeister, V. and Díaz, G. and Mosler, J.*International Journal for Numerical Methods in Engineering*117 (2019)This paper deals with two different approaches suitable for the description of plasticity in single crystals. The first one is the standard approach that is based on a continuous deformation mapping. Plasticity is driven by a classic Schmid-type relation connecting the shear stresses to the shear strains at a certain slip system. By way of contrast, the second approach is nonstandard. In this novel model, localized plastic deformation at certain slip planes is approximated by a strong discontinuity (discontinuous deformation mapping). Accordingly, a modified Schmid-type model relating the shear stresses to the shear displacements (displacement jump) is considered in this model. Although both models are indeed different, it is shown that they can be characterized by almost the same set of equations, eg, by a multiplicative decomposition of the deformation gradient into an elastic part and a plastic part. This striking analogy eventually leads to a unifying algorithmic formulation covering both models. Since the set of active slip systems is not known in advance, its determination is of utmost importance. This problem is solved here by using the nonlinear complementarity problem (NCP) as advocated by Fischer and Burmeister. While this idea is not new, it is shown that the NCP problem is well posed, independent of the number of active slip systems. To be more explicit, the tangent matrix in the return-mapping scheme is regular even for more than five simultaneously active slip systems. Based on this algorithm, texture evolution in a polycrystal is analyzed by means of both models and the results are compared in detail. © 2018 John Wiley & Sons, Ltd.view abstract 10.1002/nme.6000 **Aspects of interface elasticity theory**

Javili, A. and Ottosen, N.S. and Ristinmaa, M. and Mosler, J.*Mathematics and Mechanics of Solids*23 (2018)Interfaces significantly influence the overall material response especially when the area-to-volume ratio is large, for instance in nanocrystalline solids. A well-established and frequently applied framework suitable for modeling interfaces dates back to the pioneering work by Gurtin and Murdoch on surface elasticity theory and its generalization to interface elasticity theory. In this contribution, interface elasticity theory is revisited and different aspects of this theory are carefully examined. Two alternative formulations based on stress vectors and stress tensors are given to unify various existing approaches in this context. Focus is on the hyper-elastic mechanical behavior of such interfaces. Interface elasticity theory at finite deformation is critically reanalyzed and several subtle conclusions are highlighted. Finally, a consistent linearized interface elasticity theory is established. We propose an energetically consistent interface linear elasticity theory together with its appropriate stress measures. © 2017, The Author(s) 2017.view abstract 10.1177/1081286517699041 **On damage modeling of material interfaces: Numerical implementation and computational homogenization**

Heitbreder, T. and Ottosen, N.S. and Ristinmaa, M. and Mosler, J.*Computer Methods in Applied Mechanics and Engineering*337 (2018)A novel constitutive framework suitable for material interfaces undergoing large deformations in a geometrically exact setting was developed in Ottosen et al. (2016). In contrast to previous works, it permits the description of arbitrary material anisotropies by fulfilling all fundamental balance laws in physics as well as the principle of material objectivity. This paper deals with an efficient finite element implementation of the aforementioned framework in terms of the natural basis vectors. To be more precise, a different, more compact and more direct derivation of this framework is outlined first. It relies on the variational structure of the underlying problem. Subsequently, the aforementioned finite element approximation is elaborated which is finally embedded into a computational homogenization scheme. This scheme allows the analysis of the influence of the novel interface model on the resulting macroscopic (effective) material response. It is shown by numerical examples that the interaction of bulk energies and interface energies leads, in a very natural manner, to a complex size effect. It includes the frequently observed “the smaller the stiffer” relation, but also the less often observed “the smaller the softer” relation. However, since the overall response is usually a superposition of such relations, the effective properties cannot generally be characterized by one of the aforementioned limiting relations. © 2018 Elsevier B.V.view abstract 10.1016/j.cma.2018.03.023 **Quasi-brittle damage modeling based on incremental energy relaxation combined with a viscous-type regularization**

Langenfeld, K. and Junker, P. and Mosler, J.*Continuum Mechanics and Thermodynamics*30 (2018)This paper deals with a constitutive model suitable for the analysis of quasi-brittle damage in structures. The model is based on incremental energy relaxation combined with a viscous-type regularization. A similar approach—which also represents the inspiration for the improved model presented in this paper—was recently proposed in Junker et al. (Contin Mech Thermodyn 29(1):291–310, 2017). Within this work, the model introduced in Junker et al. (2017) is critically analyzed first. This analysis leads to an improved model which shows the same features as that in Junker et al. (2017), but which (i) eliminates unnecessary model parameters, (ii) can be better interpreted from a physics point of view, (iii) can capture a fully softened state (zero stresses), and (iv) is characterized by a very simple evolution equation. In contrast to the cited work, this evolution equation is (v) integrated fully implicitly and (vi) the resulting time-discrete evolution equation can be solved analytically providing a numerically efficient closed-form solution. It is shown that the final model is indeed well-posed (i.e., its tangent is positive definite). Explicit conditions guaranteeing this well-posedness are derived. Furthermore, by additively decomposing the stress rate into deformation- and purely time-dependent terms, the functionality of the model is explained. Illustrative numerical examples confirm the theoretical findings. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00161-018-0669-z **Variational updates for thermomechanically coupled gradient-enhanced elastoplasticity — Implementation based on hyper-dual numbers**

Fohrmeister, V. and Bartels, A. and Mosler, J.*Computer Methods in Applied Mechanics and Engineering*339 (2018)This paper deals with the implementation of thermomechanically coupled gradient-enhanced elastoplasticity at finite strains. The presented algorithmic formulation heavily relies on the variational structure of the considered initial boundary value problem. Consequently, such a variational structure is elaborated. While variational formulations are well-known in the case of isothermal plasticity theory, thermomechanically coupled gradient-enhanced plasticity theory has not been considered before. The resulting time-continuous variational principle allows the computation of all unknown variables jointly and naturally from the stationarity condition of an incremental potential. By discretization of this time-continuous potential in time, a discrete approximation is obtained which represents the foundation of the algorithmic formulation. As a matter of fact, stationarity of the respective potential again defines all unknown variables — now in a time-discrete fashion. Within this paper, the necessary condition associated with stationarity – a vanishing first gradient – is solved numerically by means of Newton's method. Hence, the first as well as the second derivatives of the incremental potential are required. They are computed by numerical differentiation based on hyper-dual numbers. By considering a perturbation with respect to hyper-dual numbers, the first as well as the second derivatives are computed in an exact manner without introducing any numerical errors. Although numerical differentiation based on hyper-dual numbers is numerically more extensive than real-valued perturbations, it is shown that the scalability of the resulting parallel finite element implementation is not dominated by this effect for sufficiently large problems. © 2018 Elsevier B.V.view abstract 10.1016/j.cma.2018.04.047 **A numerical convergence study regarding homogenization assumptions in phase field modeling**

Kiefer, B. and Furlan, T. and Mosler, J.*International Journal for Numerical Methods in Engineering*112 (2017)From a mathematical point of view, phase field theory can be understood as a smooth approximation of an underlying sharp interface problem. However, the smooth phase field approximation is not uniquely defined. Different phase field approximations are known to converge to the same sharp interface problem in the limiting case—if the thickness of the diffuse interface converges to zero. In this respect and focusing on numerics, a question that naturally arises is as follows: What are the convergence rates of the different phase field models? The paper deals precisely with this question for a certain family of phase field models. The focus is on an Allen–Cahn-type phase field model coupled to continuum mechanics. This model is rewritten into a unified variational phase field framework that covers different homogenization assumptions in the diffuse interfaces: Voigt/Taylor, Reuss/Sachs and more sound homogenization approaches falling into the range of rank-one convexification. It is shown by means of numerical experiments that the underlying phase field model—that is, the homogenization assumption in the diffuse interface—indeed influences the convergence rate. Copyright © 2017 John Wiley & Sons, Ltd. Copyright © 2017 John Wiley & Sons, Ltd.view abstract 10.1002/nme.5547 **Consistent elastoplastic cohesive zone model at finite deformations – Variational formulation**

Heitbreder, T. and Ottosen, N.S. and Ristinmaa, M. and Mosler, J.*International Journal of Solids and Structures*106-107 (2017)Cohesive zone models based on classical interface-type formulations at finite deformations are subjected to fundamental physical principles such as thermodynamical consistency, balance equations and material frame indifference. However, these restraints are often ignored and in that respect such formulations are inconsistent. By way of contrast, a consistent cohesive zone framework suitable for the analysis of localized elastoplastic deformations which only depends on the displacement jump was recently advocated in Ottosen et al. (2015). A certain subclass of this consistent framework is analyzed here, further extended and finally, an efficient numerical implementation is proposed. Conceptually, the considered cohesive zone model is a fiber-like model where the fiber direction is defined by the direction of the displacement discontinuity. A novel unloading model is advocated where the key idea is to assign a vanishing bending stiffness to the fibers and they therefore buckle when compressive stresses are initiated. Following ideas known from wrinkling in membranes, it is shown that the resulting framework can be rewritten into a variationally consistent format such that all unknowns follow jointly from minimizing a time-dependent potential whose discretization leads to an efficient implementation in terms of an efficient variational constitutive update. The physical properties of the final constitutive framework are analyzed by means of numerical examples. This analysis shows that although the framework is based on elastoplasticity, it predicts for the L-shaped structure investigated a mechanical response similar to that of damage theory even during unloading. © 2016 Elsevier Ltdview abstract 10.1016/j.ijsolstr.2016.10.027 **Efficient variational constitutive updates for Allen–Cahn-type phase field theory coupled to continuum mechanics**

Bartels, A. and Mosler, J.*Computer Methods in Applied Mechanics and Engineering*317 (2017)This paper deals with efficient variational constitutive updates for Allen–Cahn-type phase field theory coupled to a geometrically exact description of continuum mechanics. The starting point of the implementation is a unified variational principle: A time-continuous potential is introduced, the minimizers of which describe naturally every aspect of the aforementioned coupled model—including the homogenization assumptions defining the mechanical response of the bulk material in the diffuse interface region. With regard to these assumptions, classic models such as the one by Voigt/Taylor or the one by Reuss/Sachs are included. Additionally, more sound homogenization approaches falling into the range of rank-1 convexification are also included in the unified framework. Based on a direct discretization of this time-continuous potential in time and space, an efficient numerical finite element implementation is proposed. In order to guarantee admissible order parameters of the phase field, the unconstrained optimization problem is supplemented by respective constraints. They are implemented by means of Lagrange parameters combined with the Fischer–Burmeister NCP functions. This results in an exact fulfillment of the aforementioned constraints without considering any inequality. Several numerical examples show the predictive capabilities as well as the robustness and efficiency of the final algorithmic formulation. Furthermore, the influence of the homogenization assumption is analyzed in detail. It is shown that the choice of the homogenization assumption does influence the predicted microstructure in general. However, all models converge to the same solution in the limiting case. © 2016 Elsevier B.V.view abstract 10.1016/j.cma.2016.11.024 **Micro-to-macro transition accounting for general imperfect interfaces**

Javili, A. and Steinmann, P. and Mosler, J.*Computer Methods in Applied Mechanics and Engineering*317 (2017)The objective of this contribution is to establish a micro-to-macro transition framework to study the behavior of heterogeneous materials whereby the influence of interfaces at the microscale is taken into account. The term “interface” refers to a zero-thickness model that represents the finite thickness “interphase” between the constituents of the micro-structure. For geometrically equivalent samples, due to increasing area-to-volume ratio with decreasing size, interfaces demonstrate a more pronounced effect on the material response at small scales. A remarkable outcome is that including interfaces introduces a length-scale and our interface-enhanced computational homogenization captures a size effect in the material response even if linear prolongation conditions are considered. Furthermore, the interface model in this contribution is general imperfect in the sense that it allows for both jumps of the deformation as well as for the traction across the interface. Both cohesive zone model and interface elasticity theory can be derived as two limit cases of this general model. We establish a consistent computational homogenization scheme accounting for general imperfect interfaces. Suitable boundary conditions to guarantee meaningful averages are derived. Clearly, this general framework reduces to classical computational homogenization if the effect of interfaces is ignored. Finally, the proposed theory is elucidated via a series of numerical examples. © 2016 Elsevier B.V.view abstract 10.1016/j.cma.2016.12.025 **On the numerical implementation of thermomechanically coupled distortional hardening**

Bartels, A. and Mosler, J.*International Journal of Plasticity*96 (2017)In contrast to the by now classic isotropic and kinematic hardening, the more general framework of distortional hardening characterised by an evolution of the yield surface's shape can capture the effect of texture evolution on the macroscopic response. Although different distortional hardening models can indeed be found in the literature, efficient numerical implementations of such models are still missing. This statement proves particularly true within a thermomechanically coupled framework which is important for most technologically relevant processes - such as for deep drawing. Accordingly, this paper deals with an efficient finite element formulation for distortional hardening within a thermomechanically coupled framework. As a first step towards this target, the recently advocated isothermal distortional hardening framework Shi et al. (2014) is extended to the thermomechanically coupled setting. In order to avoid an over-estimation of the temperature increase due to plastic deformation, the initial yield stress is decomposed into a classic dissipative part and a non-classic energetic part. By doing so, the restrictions imposed by thermodynamical principles are fulfilled and simultaneously realistic temperature predictions are obtained. For the resulting model, an efficient numerical implementation is proposed. By developing a suitable time integration scheme for the evolution equations of the fourth-order tensor describing the distortional hardening, a return-mapping scheme for updating the internal variables is derived which shows the same numerical complexity as a return-mapping scheme for purely isotropic hardening. This efficient return-mapping scheme is finally incorporated into a thermomechanically coupled finite element formulation, and the resulting set of equations is fully implicitly and monolithically solved by means of a Newton-type iteration. Several numerical complex examples demonstrate the capabilities of the distortional hardening model as well as the robustness and efficiency of the numerical formulation. © 2017 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijplas.2017.05.003 **A variational formulation for thermomechanically coupled low cycle fatigue at finite strains**

Canadija, M. and Mosler, J.*International Journal of Solids and Structures*100-101 (2016)In this paper, a constitutive model suitable for the analysis of Low Cycle Thermo-Mechanical Fatigue in metals is elaborated. The model is based on finite strain elastoplasticity coupled to continuum damage theory. It is embedded into a thermodynamical framework allowing to consistently capture the interplay between mechanics and thermal effects. It is shown that the fully coupled constitutive model can be re-written into a variationally consistent manner such that all (state) variables follow jointly and naturally from minimizing an incrementally defined functional. By discretizing this time-continuous functional in time by means of implicit integration schemes a numerically efficient implementation is proposed. In order to predict the temperature increase caused by plastic deformations realistically, the pre-loading history of the considered specimen is accounted for by non-zero initial internal variables. A comparison of the results predicted by the novel constitutive model to those corresponding to experiments (Ultimet alloy) shows that the predictive capabilities of the final model are excellent. © 2016view abstract 10.1016/j.ijsolstr.2016.09.009 **Framework for non-coherent interface models at finite displacement jumps and finite strains**

Ottosen, N.S. and Ristinmaa, M. and Mosler, J.*Journal of the Mechanics and Physics of Solids*90 (2016)This paper deals with a novel constitutive framework suitable for non-coherent interfaces, such as cracks, undergoing large deformations in a geometrically exact setting. For this type of interface, the displacement field shows a jump across the interface. Within the engineering community, so-called cohesive zone models are frequently applied in order to describe non-coherent interfaces. However, for existing models to comply with the restrictions imposed by (a) thermodynamical consistency (e.g., the second law of thermodynamics), (b) balance equations (in particular, balance of angular momentum) and (c) material frame indifference, these models are essentially fiber models, i.e. models where the traction vector is collinear with the displacement jump. This constraints the ability to model shear and, in addition, anisotropic effects are excluded. A novel, extended constitutive framework which is consistent with the above mentioned fundamental physical principles is elaborated in this paper. In addition to the classical tractions associated with a cohesive zone model, the main idea is to consider additional tractions related to membrane-like forces and out-of-plane shear forces acting within the interface. For zero displacement jump, i.e. coherent interfaces, this framework degenerates to existing formulations presented in the literature. For hyperelasticity, the Helmholtz energy of the proposed novel framework depends on the displacement jump as well as on the tangent vectors of the interface with respect to the current configuration - or equivalently - the Helmholtz energy depends on the displacement jump and the surface deformation gradient. It turns out that by defining the Helmholtz energy in terms of the invariants of these variables, all above-mentioned fundamental physical principles are automatically fulfilled. Extensions of the novel framework necessary for material degradation (damage) and plasticity are also covered. © 2016 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.jmps.2016.02.034 **Fundamental physical principles and cohesive zone models at finite displacements - Limitations and possibilities**

Ottosen, N.S. and Ristinmaa, M. and Mosler, J.*International Journal of Solids and Structures*53 (2015)Cohesive zone modeling at finite displacements and crack openings are discussed and evaluated in relation to the fundamental principles provided by thermodynamic consistency, balance of angular momentum and material frame indifference. It is shown that among the existing prototype cohesive zone models within the classical cohesive zone framework, only fully isotropic elastic models or fully isotropic elastic models combined with isotropic damage theory fulfill all of the aforementioned fundamental principles whereas present elasto-plastic models do not. For this reason, and still within the classical cohesive zone framework, a novel elasto-plastic model is proposed which fulfills all requirements. Subsequently, the classical cohesive zone framework is generalized and within this new framework a consistent elasto-plastic model is proposed. © 2014 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijsolstr.2014.10.020 **Large deformation framework for phase-field simulations at the mesoscale**

Borukhovich, E. and Engels, P.S. and Mosler, J. and Shchyglo, O. and Steinbach, I.*Computational Materials Science*108 (2015)Abstract A large-strain plasticity framework is set up for phase-field simulations at the mesoscopic scale. The approach is based on an Eulerian setting with remeshing after each time step to keep a fixed structured mesh. Rotations, as evaluated from the antisymmetric part of the deformation gradient tensor, are integrated to capture the process history. Special emphasis is also given to the homogenization of the diffuse interface region to ensure the Hadamard jump condition and 2-dimensional scaling of the interface. The approach is applied to deformation of a polycrystal. © 2015 Elsevier B.V.view abstract 10.1016/j.commatsci.2015.06.021 **On the thermomechanical coupling in dissipative materials: A variational approach for generalized standard materials**

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

Mosler, J. and Shchyglo, O. and Montazer Hojjat, H.*Journal of the Mechanics and Physics of Solids*68 (2014)This paper deals with the analysis of homogenization assumptions within phase field theories in a finite strain setting. Such homogenization assumptions define the average bulks energy within the diffusive interface region where more than one phase co-exist. From a physical point of view, a correct computation of these energies is essential, since they define the driving force of material interfaces between different phases. The three homogenization assumptions considered in this paper are: (a) Voigt/Taylor model, (b) Reuss/Sachs model, and (c) Khachaturyan model. It is shown that these assumptions indeed share some similarities and sometimes lead to the same results. However, they are not equivalent. Only two of them allow the computation of the individual energies of the co-existing phases even within the aforementioned diffusive interface region: the Voigt/Taylor and the Reuss/Sachs model. Such a localization of the averaged energy is important in order to determine and to subsequently interpret the driving force at the interface. Since the Voigt/Taylor and the Reuss/Sachs model are known to be relatively restrictive in terms of kinematics (Voigt/Taylor) and linear momentum (Reuss/Sachs), a novel homogenization approach is advocated. Within a variational setting based on (incremental) energy minimization, the results predicted by the novel approach are bounded by those corresponding to the Voigt/Taylor and the Reuss/Sachs model. The new approach fulfills equilibrium at material interfaces (continuity of the stress vector) and it is kinematically compatible. In sharp contrast to existing approaches, it naturally defines the mismatch energy at incoherent material interfaces. From a mathematical point of view, it can be interpreted as a partial rank-one convexification. © 2014 Elsevier Ltd.view abstract 10.1016/j.jmps.2014.04.002 **On the thermodynamically consistent modeling of distortional hardening: A novel generalized framework**

Shi, B. and Bartels, A. and Mosler, J.*International Journal of Plasticity*63 (2014)Many important physical effects of materials undergoing plasticity at the macroscale cannot be captured realistically by isotropic and kinematic hardening only. For instance, the evolution of the texture in polycrystals results macroscopically in a distorted yield surface. This paper deals with adequate hardening models for such a distortion. To be more precise, a novel general frame for finite strain plasticity models is elaborated. To the best knowledge of the authors, it is the first one combining the following features: (1) proof of thermodynamical consistency; (2) decomposition of distortional hardening into dynamic hardening (due to currently active dislocations) and latent hardening (due to currently inactive dislocations); (3) difference of the yield surface's curvature in loading direction and in the opposite direction. The cornerstone of this model is a new plastic potential for the evolution equations governing distortional hardening. Although this type of hardening is characterized through a fourth-order tensor as internal variable, the structure of the aforementioned potential is surprisingly simple. Even though the final model is rather complex, it requires only few model parameters. For these parameters, in turn, physically sound bounds based on the convexity condition of the yield surface can be derived. Three different examples demonstrate the predictive capabilities of the novel framework. © 2014 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijplas.2014.05.008 **A hybrid variationally consistent homogenization approach based on Ritz's method**

Bleier, N. and Mosler, J.*International Journal for Numerical Methods in Engineering*94 (2013)Multiscale approaches based on homogenization theory provide a suitable framework to incorporate information associated with a small-scale (microscale) problem into the considered large-scale (macroscopic) problem. In this connection, the present paper proposes a novel computationally efficient hybrid homogenization method. Its backbone is a variationally consistent FE2 approach in which every aspect is governed by energy minimization. In particular, scale bridging is realized by the canonical principle of energy equivalence. As a direct implementation of the aforementioned variationally consistent FE2 approach is numerically extensive, an efficient approximation based on Ritz's method is advocated. By doing so, the material parameters defining an effective macroscopic material model capturing the underlying microstructure can be efficiently computed. Furthermore, the variational scale bridging principle provides some guidance to choose a suitable family of macroscopic material models. Comparisons between the results predicted by the novel hybrid homogenization method and full field finite element simulations show that the novel method is indeed very promising for multiscale analyses. © 2013 John Wiley & Sons, Ltd.view abstract 10.1002/nme.4465 **Experimental and numerical investigation of Mg alloy sheet formability**

Mekonen, M.N. and Steglich, D. and Bohlen, J. and Stutz, L. and Letzig, D. and Mosler, J.*Materials Science and Engineering A*586 (2013)The current paper explores experimentally and numerically obtained mechanical responses of the Nakazima-type sheet forming for the magnesium alloys ZE10 and AZ31 at elevated temperature (200. °C). The results from the experiments revealed sufficient ductility allowing sheet forming processes at the prescribed test temperature. The material's anisotropy recorded in previous experiments was confirmed. Differences in the mechanical response between the two materials in terms of strain paths during the forming experiments were quantified. The corresponding numerical responses were obtained employing a suitable constitutive model taking into account the characteristic anisotropy in deformation. In addition, for predicting limit conditions of the forming process, the localization criterion by Marciniak and Kuczynski was adopted. The constitutive model together with the localization criterion was implemented in a finite element framework based on a fully implicit time integration scheme. The reasonably good agreement between the responses of the model and the respective experiments indicated the predictive capabilities of the implemented model for the considered magnesium alloys. © 2013 Elsevier B.V.view abstract 10.1016/j.msea.2013.07.088 **On the interaction between different size effects in fibre reinforced PMMA: Towards composites with optimised fracture behaviour**

Scheider, I. and Xiao, T. and Huber, N. and Mosler, J.*Computational Materials Science*80 (2013)This paper is concerned with a numerical investigation of different size effects and their interactions in fibre reinforced PMMA. Focus is on the mechanical response - particularly on the damage and the fracture behaviour. The performed numerical studies are based on finite element simulations in which representative volume elements with different microstructures have been virtually mechanically tested and compared to each other. The underlying numerical model captures the most relevant mechanical mechanisms such as damage evolution, crack propagation and failure by a cohesive zone model. Previous studies have shown that the effective macroscopic fracture properties can be changed by varying the thickness of the fibres. In this paper, an additional size effect resulting from a variation of the fibres' lengths and the interaction between both size effects is carefully analysed. By understanding such size effects, the observed failure mechanisms can be changed effectively and the properties of the considered composite can be improved significantly. For instance, it will be shown that a composite can be designed which shows a high strength as well as a high fracture energy. © 2013 Elsevier B.V. All rights reserved.view abstract 10.1016/j.commatsci.2013.04.027 **On the macroscopic description of yield surface evolution by means of distortional hardening models: Application to magnesium**

Shi, B. and Mosler, J.*International Journal of Plasticity*44 (2013)Texture evolution in polycrystals due to rotation of the atomic lattice in single grains results in a complex macroscopic mechanical behavior which cannot be reasonably captured only by classical isotropic or kinematic hardening in general. More precisely and focusing on standard rate-independent plasticity theory, the complex interplay at the microscale of a polycrystal leads to an evolving macroscopic anisotropy of the yield surface, also known as distortional or differential hardening. This effect is of utmost importance, if non-radial loading paths such as those associated with forming processes are to be numerically analyzed. In the present paper, different existing distortional hardening models are critically reviewed. For a better comparison, they are reformulated into the modern framework of hyperelastoplasticity, and the same objective time derivative is applied to all evolution equations. Furthermore, since the original models are based on a Hill-type yield function not accounting for the strength differential effect as observed in hcp metals such as magnesium, respective generalizations are also discussed. It is shown that only one of the resulting models can fulfill the second law of thermodynamics. That model predicts a high curvature of the yield function in loading direction, while the opposite region of the yield function is rather flat. Indeed, such a response can be observed for some materials such as aluminum alloys. In the case of magnesium, however, that does not seem to be true. Therefore, a novel constitutive model is presented. Its underlying structure is comparably simple and the model is thermodynamically consistent. Conceptually, distortional hardening is described by an Armstrong-Frederick-type evolution equation. The predictive capabilities of the final model are demonstrated by comparisons to experimentally measured data. © 2013 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijplas.2012.11.007 **Size effects in short fibre reinforced composites**

Scheider, I. and Chen, Y. and Hinz, A. and Huber, N. and Mosler, J.*Engineering Fracture Mechanics*100 (2013)The present paper is concerned with the analysis of size effects in short fibre reinforced composites. The microstructure of such composites often represents the first hierarchy level of a bioinspired material. For modelling fibre cracking as well as debonding between fibre and matrix material, a fully three-dimensional cohesive zone model is applied. It is shown that this model indeed captures the size effect associated with material failure of a single fibre. Furthermore, this scaling effect strongly depends on the shape and orientation of the assumed pre-existing crack. For this reason, a two-dimensional description can usually only predict the size effect qualitatively. Based on the aforementioned findings, a representative volume element (RVE) containing ceramic fibres embedded within a polymer matrix is considered. Similar to the single fibre, the RVE also shows a pronounced size effect. However, the underlying physical process is significantly more complex. More explicitly, the size effect of the RVE is a superposition of that related to the isolated fibres as well as of that induced by debonding of the fibres from the matrix material. For estimating the different effects, a perfect bond is also modelled. © 2012 Elsevier Ltd.view abstract 10.1016/j.engfracmech.2012.05.005 **Efficient modeling of microstructure evolution in magnesium by energy minimization**

Homayonifar, M. and Mosler, J.*International Journal of Plasticity*28 (2012)The description of the complex interplay between deformation-induced twinning and dislocation slip, typical for metals showing an hcp structure such as magnesium, is of utmost importance for understanding their deformation behavior. In the present paper, an incremental energy principle is presented for that purpose. Within this principle, dislocation slip is modeled by crystal plasticity theory, while the phase decomposition associated with twinning is considered by a mixture theory. This mixture theory naturally avoids the decomposition of the twinning process into so-called pseudo-dislocations followed by a reorientation of the total crystal. By way of contrast, the proposed model captures the transformation of the crystal lattice due to twinning in a continuous fashion by simultaneously taking dislocation slip within both, possibly co-existent, phases into account. The shear strain induced by twinning as well as the deformation history are consistently included within the twinned domain by an enhanced multiplicative decomposition of the deformation gradient. Kinematic compatibility between the different phases is enforced by a Hadamard-type compatibility condition, while compatibility with respect to the boundary conditions requires the introduction of a boundary layer. The evolution of all state variables such as the twinning volume and the plastic strains associated with dislocation slip follow jointly and conveniently from minimizing the stress power of the total crystal. This canonical variational principle is closely related to the postulate of maximum dissipation and guarantees thermodynamical consistency of the resulting model. Particularly, the second law of thermodynamics is fulfilled. In contrast to previous models suitable for the analysis of the deformation systems in magnesium, the Helmholtz energy of the twinning interfaces and that of the aforementioned boundary layer are considered. Analogously, the energy due to twinning nucleation and that related to twinning growth are accounted for by suitable dissipation functionals. By doing so, the number of twinning laminates becomes an additional unknown within the minimization principle and thus, the thickness of the lamellas can be computed. Interestingly, by interpreting this thickness as the mean free path of dislocations, a size effect of Hall-Petch-type can naturally be included within the novel model. The predictive capabilities of the resulting approach are finally demonstrated by analyzing the channel die test. For that purpose, a certain rank-two laminate structure is considered. However, it bears emphasis that the proposed framework is very general and consequently, it can also be applied to other materials. © 2011 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijplas.2011.05.011 **Efficient variational constitutive updates by means of a novel parameterization of the flow rule**

Bleier, N. and Mosler, J.*International Journal for Numerical Methods in Engineering*89 (2012)Analogously to the classical return-mapping algorithm, so-called variational constitutive updates are numerical methods allowing to compute the unknown state variables such as the plastic strains and the stresses for material models showing an irreversible mechanical response. In sharp contrast to standard approaches in computational inelasticity, the state variables follow naturally and jointly from energy minimization in case of variational constitutive updates. This leads to significant advantages from a numerical, mathematical as well as from a physical point of view. However, while the classical return-mapping algorithm has been being developed for several decades, and thus, it has already reached a certain maturity, variational constitutive updates have drawn attention only relatively recently. This is particularly manifested in the numerical performance of such algorithms. Within the present paper, the numerical efficiency of variational constitutive updates is critically analyzed. It will be shown that a naive approximation of the flow rule causes a singular Hessian within the respective Newton-Raphson scheme. However, by developing a novel parameterization of the flow rule, an efficient algorithm is derived. Its performance is carefully compared to that of the classical return-mapping scheme. This comparison clearly shows that the novel variationally consistent implementation is, at least, as efficient as the classical return-mapping algorithm. © 2011 John Wiley & Sons, Ltd.view abstract 10.1002/nme.3280 **Experimental and numerical lifetime assessment of Al 2024 sheet**

Khan, S. and Kintzel, O. and Mosler, J.*International Journal of Fatigue*37 (2012)In the present paper, a thorough analysis of the low-cycle fatigue behavior of flat sheets of aluminum Al 2024-T351 is given. For that purpose, material characterization is combined with material modeling. The experimental analyses include monotonic and cyclic loading tests at high stress levels. For the assessment of microstructural characteristics, advanced imaging technology is used to reveal, e.g. crack initiation loci and particle sizing. The numerical simulation is done using a novel ductile-brittle damage model. Thereby, the model parameters are optimized by means of an inverse parameter identification strategy which, overall, leads to a very good agreement between experimentally observed and computationally predicted data. For demonstrating the prediction capability of the novel coupled model also for complex engineering problems, a certain stringer assembly, as used in fuselage parts of airplanes, is analyzed. © 2011 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijfatigue.2011.09.010 **Low cycle fatigue damage mechanism of the lightweight alloy Al2024**

Khan, S. and Wilde, F. and Beckmann, F. and Mosler, J.*International Journal of Fatigue*38 (2012)Detection of cracks in Al2024 T351 specimens subjected to low cycle fatigue loading by a certain non-destructive inspection technique is demonstrated. In the experimental phase of the study, notched round specimens were fatigue loaded. The tests were performed at different constant strain amplitudes at room temperature. For identifying the crack initiation loci, the specimens were removed from the testing machine after a certain number of cycles and were non-destructively inspected via X-ray technique. Pictures were taken successively while incrementally turning the sample. The re-constructed data were visualized via software (VGStudio MAX 2.1) to obtain a 3D image of the specimen, showing all the details of its inner structure. By taking "virtual" slices from the data, quantification of microstructural properties was done using classical methods. This allowed verifying some frequently mentioned statements concerning the low cycle fatigue behavior of high-strength aluminum alloys. Furthermore, new findings related to the tri-axiality dependence on the resulting fracture process and those related to damage initiation caused by decohesion were also discovered. © 2011 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijfatigue.2011.11.009 **Mechanical characterization and constitutive modeling of Mg alloy sheets**

Mekonen, M.N. and Steglich, D. and Bohlen, J. and Letzig, D. and Mosler, J.*Materials Science and Engineering A*540 (2012)In this paper, an experimental mechanical characterization of the magnesium alloys ZE10 and AZ31 is performed and a suitable constitutive model is established. The mechanical characterization is based on uniaxial tensile tests. In order to avoid poor formability at room temperature, the tests were conducted at elevated temperature (200. °C). The uniaxial tensile tests reveal sufficient ductility allowing sheet forming processes at this temperature. The differences in yield stresses and plastic strain ratios (r-values) confirm the anisotropic response of the materials under study. The constitutive model is established so that the characteristic mechanical features observed in magnesium alloys such as anisotropy and compression-tension asymmetry can be accommodated. This model is thermodynamically consistent, incorporates rate effect, is formulated based on finite strain plasticity theory and is applicable in sheet forming simulations of magnesium alloys. More precisely, a model originally proposed by Cazacu and Barlat in 2004 and later modified to account for the evolution of the material anisotropy is rewritten in a thermodynamically consistent framework. The calibrated constitutive model is shown to capture the characteristic mechanical features observed in magnesium alloy sheets. © 2012 Elsevier B.V.view abstract 10.1016/j.msea.2012.01.122 **Variational constitutive updates for microstructure evolution in hcp metals**

Mosler, J. and Homayonifar, M.*GAMM Mitteilungen*35 (2012)Magnesium and its alloys are promising materials for lightweight applications. Unfortunately, the macroscopic formability of such materials is relatively poor at room temperature and these metals are characterized by a complex mechanical response. This response is a result of the interplay between different deformation modes at the microscale. Since magnesium is a material showing a hexagonal close-packed (hcp) structure of the underlying atomic lattice, plasticity caused by dislocations and deformation-induced twinning are the most relevant deformation modes. Within the present paper, two different recently advocated modeling approaches suitable for capturing such modes at the microscale are analyzed. It is shown that both models can be rewritten into a variationally consistent format where every aspect is naturally driven by energy minimization. In addition to this already known feature, it turns out that both models are based on the same minimization problem. The difference between the models results from different constraints enforced within the variational principle. For getting further insight into the interaction between dislocations and twinning interfaces, accompanying atomistic simulations based on molecular dynamics are also performed. The results of such simulations enter the micromechanical model through the initial plastic deformation within the twinned phase. ©c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201210004 **A thermodynamically and variationally consistent class of damage-type cohesive models**

Mosler, J. and Scheider, I.*Journal of the Mechanics and Physics of Solids*59 (2011)A novel class of cohesive constitutive models suitable for the analysis of material separation such as that related to cracks, shear bands or delamination processes is presented. The proposed framework is based on a geometrically exact description (finite deformation) and it naturally accounts for material anisotropies. For that purpose, a Helmholtz energy depending on evolving structural tensors is introduced. In sharp contrast to previously published anisotropic cohesive models with finite strain kinematics based on a spatial description, all models belonging to the advocated class are thermodynamically consistent, i.e., they are rigorously derived by applying the Coleman and Noll procedure. Although this procedure seems nowadays to be standard for stressstrain-type constitutive laws, this is not the case for cohesive models at finite strains. An interesting new finding from the Coleman and Noll procedure is the striking analogy between cohesive models and boundary potential energies. This analogy gives rise to the introduction of additional stress tensors which can be interpreted as deformational surface shear. To the best knowledge of the authors, those stresses which are required for thermodynamical consistency at finite strains, have not been taken into account in existing models yet. Furthermore, the additional stress tensors can result in an effective traction-separation law showing a non-trivial stress-free configuration consistent with the underlying Helmholtz energy. This configuration is not predicted by previous models. Finally, the analogy between cohesive models and boundary potential energies leads to a unique definition of the controversially discussed fictitious intermediate configuration. More precisely, traction continuity requires that the interface geometry with respect to the deformed configuration has to be taken as the average of both sides. It will be shown that the novel class of interface models does not only fulfill the second law of thermodynamics, but also it shows an even stronger variational structure, i.e., the admissible states implied by the novel model can be interpreted as stable energy minimizers. This variational structure is used for deriving a variationally consistent numerical implementation. © 2011 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.jmps.2011.04.012 **An efficient algorithm for the inverse problem in elasticity imaging by means of variational r-adaption**

Arnold, A. and Bruhns, O.T. and Mosler, J.*Physics in Medicine and Biology*56 (2011)A novel finite element formulation suitable for computing efficiently the stiffness distribution in soft biological tissue is presented in this paper. For that purpose, the inverse problem of finite strain hyperelasticity is considered and solved iteratively. In line with Arnold et al (2010 Phys. Med. Biol. 55 2035), the computing time is effectively reduced by using adaptive finite element methods. In sharp contrast to previous approaches, the novel mesh adaption relies on an r-adaption (re-allocation of the nodes within the finite element triangulation). This method allows the detection of material interfaces between healthy and diseased tissue in a very effective manner. The evolution of the nodal positions is canonically driven by the same minimization principle characterizing the inverse problem of hyperelasticity. Consequently, the proposed mesh adaption is variationally consistent. Furthermore, it guarantees that the quality of the numerical solution is improved. Since the proposed r-adaption requires only a relatively coarse triangulation for detecting material interfaces, the underlying finite element spaces are usually not rich enough for predicting the deformation field sufficiently accurately (the forward problem). For this reason, the novel variational r-refinement is combined with the variational h-adaption (Arnold et al 2010) to obtain a variational hr-refinement algorithm. The resulting approach captures material interfaces well (by using r-adaption) and predicts a deformation field in good agreement with that observed experimentally (by using h-adaption). © 2011 Institute of Physics and Engineering in Medicine.view abstract 10.1088/0031-9155/56/14/004 **An incremental minimization principle suitable for the analysis of low cycle fatigue in metals: A coupled ductile-brittle damage model**

Kintzel, O. and Mosler, J.*Computer Methods in Applied Mechanics and Engineering*200 (2011)The present paper is concerned with a novel variational constitutive update suitable for the analysis of low cycle fatigue in metals. The underlying constitutive model originally advocated in [1] accounts for plastic deformation as well as for damage accumulation. The latter is captured by a combination of two constitutive models. While the first of those is associated with ductile damage, the second material law is related to a quasi-brittle response. The complex overall model falls into the range of so-called generalized standard materials and thus, it is thermodynamically consistent. However, since the evolution equations are non-associative, it does not show an obvious variational structure. By enforcing the flow rule as well as the evolution equations through a suitable parameterization, a minimization principle can be derived nevertheless. Discretized in time, this principle is employed for developing an effective numerical implementation. Since the mechanical subproblems corresponding to ductile damage and that of quasi-brittle damage are uncoupled, an efficient staggered scheme can be elaborated. Within both steps, Newton's method is applied. While the evolution of the quasi-brittle damage requires only the computation of a one-dimensional optimization problem, the ductile damage model is defined by a numerically more expensive tensor-valued variable. For further increasing the numerical performance of the respective minimization principle, a closed-form solution for the inverse of the Hessian matrix is derived. By numerically analyzing the prediction of mesocrack initiation in low-cycle fatigue simulations, the performance of the resulting algorithm is demonstrated. © 2011 Elsevier B.V.view abstract 10.1016/j.cma.2011.07.006 **Effective 3D failure simulations by combining the advantages of embedded Strong Discontinuity Approaches and classical interface elements**

Radulovic, R. and Bruhns, O.T. and Mosler, J.*Engineering Fracture Mechanics*78 (2011)An efficient numerical framework suitable for three-dimensional analyses of brittle material failure is presented. The proposed model is based on an (embedded) Strong Discontinuity Approach (SDA). Hence, the deformation mapping is elementwise additively decomposed into a conforming, continuous part and an enhanced part associated with the kinematics induced by material failure. To overcome locking effects and to provide a continuous crack path approximation, the approach is extended and combined with advantages known from classical interface elements. More precisely, several discontinuities each of them being parallel to a facet of the respective finite element are considered. By doing so, crack path continuity is automatically fulfilled and no tracking algorithm is necessary. However, though this idea is similar to that of interface elements, the novel SDA is strictly local (finite element level) and thus, it does not require any modification of the global data structure, e.g. no duplication of nodes. An additional positive feature of the advocated finite element formulation is that it leads to a symmetric tangent matrix. It is shown that several simultaneously active discontinuities in each finite element are required to capture highly localized material failure. The performance and predictive ability of the model are demonstrated by means of two benchmark examples. © 2011 Elsevier Ltd.view abstract 10.1016/j.engfracmech.2011.06.007 **Efficient modeling of localized material failure by means of a variationally consistent embedded strong discontinuity approach**

Mosler, J. and Stanković, L. and Radulović, R.*International Journal for Numerical Methods in Engineering*88 (2011)This paper is concerned with a novel embedded strong discontinuity approach suitable for the analysis of material failure at finite strains. Focus is on localized plastic deformation particularly relevant for slip bands. In contrast to already existing models, the proposed implementation allows to consider several interacting discontinuities in each finite element. Based on a proper re-formulation of the kinematics, an efficient parameterization of the deformation gradient is derived. It permits to compute the strains explicitly that improves the performance significantly. However, the most important novel contribution of the present paper is the advocated variational constitutive update. Within this framework, every aspect is naturally driven by energy minimization, i.e. all unknown variables are jointly computed by minimizing the stress power. The proposed update relies strongly on an extended principle of maximum dissipation. This framework provides enough flexibility for different failure types and for a broad class of non-associative evolution equations. By discretizing the aforementioned continuous variational principle, an efficient numerical implementation is obtained. It shows, in addition to its physical and mathematical elegance, several practical advantages. For instance, the physical minimization principle itself specifies automatically and naturally the set of active strong discontinuities. © 2011 John Wiley & Sons, Ltd.view abstract 10.1002/nme.3210 **Novel approach for the treatment of cyclic loading using a potential-based cohesive zone model**

Scheider, I. and Mosler, J.*Procedia Engineering*10 (2011)The development of cohesive zone models in the finite element framework dates back some 30 years, and cohesive interface elements are nowadays employed as a standard tool in scientific and engineering communities. They have been successfully applied to a broad variety of different materials and loading scenarios. However, many of such constitutive models are simply based on traction-separation relations without deducing them from energy potentials. By way of contrast, a thermodynamically consistent cohesive zone model suitable for the analysis of low cycle fatigue is elaborated in the present contribution. For that purpose, a plasticity-based cohesive law including isotropic hardening/softening is supplemented by a damage model. First results of this new approach to cyclic loading will be presented illustrating the applicability to low cycle fatigue. © 2011 Published by Elsevier Ltd.view abstract 10.1016/j.proeng.2011.04.358 **Numerical simulation of forming limit test for AZ31 at 200°C**

Nebebe, M. and Bohlen, J. and Steglich, D. and Mosler, J.*Key Engineering Materials*473 (2011)This work is concerned with numerical analyses of the forming behavior of magnesium at elevated temperature. For that purpose, a thermodynamically consistent, rate-dependent, finite-strain elasto-plastic constitutive model is presented. This model captures the stress differential effect as well as the anisotropy of magnesium. Furthermore, the change in shape of the yield locus (distortional hardening) is also taken into account. This constitutive law, together with its parameter calibration based on uni-axial tensile tests, is finally combined with the localization criterion originally proposed by Marciniak and Kuczynski and applied to the simulation of forming limit test. Comparisons to experiments show the excellent predictive capabilities of the model. © (2011) Trans Tech Publications.view abstract 10.4028/www.scientific.net/KEM.473.468 **On the coupling of plastic slip and deformation-induced twinning in magnesium: A variationally consistent approach based on energy minimization**

Homayonifar, M. and Mosler, J.*International Journal of Plasticity*27 (2011)The present paper is concerned with the analysis of the deformation systems in single crystal magnesium at the micro-scale and with the resulting texture evolution in a polycrystal representing the macroscopic mechanical response. For that purpose, a variationally consistent approach based on energy minimization is proposed. It is suitable for the modeling of crystal plasticity at finite strains including the phase transition associated with deformation-induced twinning. The method relies strongly on the variational structure of crystal plasticity theory, i.e.; an incremental minimization principle can be derived which allows to determine the unknown slip rates by computing the stationarity conditions of a (pseudo) potential. Phase transition associated with twinning is modeled in a similar fashion. More precisely, a solid-solid phase transition corresponding to twinning is assumed, if this is energetically favorable. Mathematically speaking, the aforementioned transition can be interpreted as a certain rank-one convexification. Since such a scheme is computationally very expensive and thus, it cannot be applied to the analysis of a polycrystal, a computationally more efficient approximation is elaborated. Within this approximation, the deformation induced by twinning is decomposed into the reorientation of the crystal lattice and simple shear. The latter is assumed to be governed by means of a standard Schmid-type plasticity law (pseudo-dislocation), while the reorientation of the crystal lattice is considered, when the respective plastic shear strain reaches a certain threshold value. The underlying idea is in line with experimental observations, where dislocation slip within the twinned domain is most frequently seen, if the twin laminate reaches a critical volume. The resulting model predicts a stress-strain response in good agreement with that of a rank-one convexification method, while showing the same numerical efficiency as a classical Taylor-type approximation. Consequently, it combines the advantages of both limiting cases. The model is calibrated for single crystal magnesium by means of the channel die test and finally applied to the analysis of texture evolution in a polycrystal. Comparisons of the predicted numerical results to their experimental counterparts show that the novel model is able to capture the characteristic mechanical response of magnesium very well. © 2010 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijplas.2010.10.009 **On the thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening by means of incremental energy minimization**

Canadija, M. and Mosler, J.*International Journal of Solids and Structures*48 (2011)The thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening is analyzed within the present paper. This coupling is of utmost importance in many applications, e.g., in those showing low cycle fatigue (LCF) under large strain amplitudes. Since the by now classical thermomechanical coupling originally proposed by Taylor and Quinney cannot be used directly in case of kinematic hardening, the change in heat as a result of plastic deformation is computed by applying the first law of thermodynamics. Based on this balance law, together with a finite strain plasticity model, a novel variationally consistent method is elaborated. Within this method and following Stainier and Ortiz (2010), all unknown variables are jointly and conveniently computed by minimizing an incrementally defined potential. In sharp contrast to previously published works, the evolution equations are a priori enforced by employing a suitable parameterization of the flow rule and the evolution equations. The advantages of this parameterization are, at least, twofold. First, it leads eventually to an unconstrained stationarity problem which can be directly applied to any yield function being positively homogeneous of degree one, i.e., the approach shows a broad range of application. Secondly, the parameterization provides enough flexibility even for a broad range of non-associative models such as kinematic hardening of Armstrong-Frederick-type. Different to Stainier and Ortiz (2010), the continuous variational problem is approximated by a standard, fully-implicit time integration. The applicability of the resulting numerical implementation is finally demonstrated by analyzing the thermodynamically coupled response for a loading cycle. © 2011 Elsevier Ltd.view abstract 10.1016/j.ijsolstr.2010.12.018 **A novel isotropic quasi-brittle damage model applied to LCF analyses of Al2024**

Kintzel, O. and Khan, S. and Mosler, J.*International Journal of Fatigue*32 (2010)The current paper deals with the assessment and the numerical simulation of low cycle fatigue of an aluminum 2024 alloy. According to experimental observations, the material response of Al2024 is highly direction-dependent showing a material behavior between ductile and brittle. In particular, in its corresponding (small transversal) S-direction, the material behavior can be characterized as quasi-brittle. For the modeling of such a mechanical response, a novel, fully coupled isotropic ductile-brittle continuum damage mechanics model is proposed. Since the resulting model shows a large number of material parameters, an efficient, hybrid parameter identification strategy is discussed. Within this strategy, as many parameters as possible have been determined a priori by exploiting analogies to established theories (like Paris' law), while the remaining free unknowns are computed by solving an optimization problem. Comparisons between the experimentally observed and the numerically simulated lifetimes reveal the prediction capability of the proposed model. © 2010 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijfatigue.2010.07.001 **An experimental and numerical lifetime assessment of Al 2024 sheet**

Saleem Khan, S. and Hellmann, D. and Kintzel, O. and Mosler, J.*Procedia Engineering*2 (2010)This work is concerned with the simulation of ductile damage evolution and the respective final material failure resulting from load reversals (LCF). For that purpose, a continuum damage mechanics (CDM) model proposed by Lemaitre is modified and utilized. This model has been validated by means of experiments of Al2024 alloy. These experiments involve specimens with different testing conditions. First, monotonic tensile tests have been considered. Subsequently, the cyclic yielding behavior has been characterized performing cyclic plasticity and damage tests on flat smooth specimen. The predictions of the model are compared to the experimentally observed results. Within the present work, special emphasis is placed on the experimental setup for fatigue testing of flat specimens as well as on the predictions of the number of cycles to final failure and the crack initiation loci. © 2010 Published by Elsevier Ltd.view abstract 10.1016/j.proeng.2010.03.123 **An investigation on low cycle lifetime of AL2024 alloy**

Vyshnevskyy, A. and Khan, S. and Mosler, J.*Key Engineering Materials*417-418 (2010)One of the important considerations in the design of components is the estimation of cyclic lifetime and analysis of the critical regions of a construction. The local approach of lifetime estimation using continuum damage mechanics (CDM) has shown a great potential in predicting material failure not only for monotonic, but also for fully reversed loadings. In this paper, the CDM model of Desmorat-Lemaitre [1] was investigated regarding the prediction of cyclic lifetime. A series of experiments on tension specimens with different geometries were performed. The latter were used for the determination of model parameters as well as for the validation of the predictive capability of the model. © (2010) Trans Tech Publications, Switzerland.view abstract 10.4028/www.scientific.net/KEM.417-418.289 **Efficient computation of the elastography inverse problem by combining variational mesh adaption and a clustering technique**

Arnold, A. and Reichling, S. and Bruhns, O.T. and Mosler, J.*Physics in Medicine and Biology*55 (2010)This paper is concerned with an efficient implementation suitable for the elastography inverse problem. More precisely, the novel algorithm allows us to compute the unknown stiffness distribution in soft tissue by means of the measured displacement field by considerably reducing the numerical cost compared to previous approaches. This is realized by combining and further elaborating variational mesh adaption with a clustering technique similar to those known from digital image compression. Within the variational mesh adaption, the underlying finite element discretization is only locally refined if this leads to a considerable improvement of the numerical solution. Additionally, the numerical complexity is reduced by the aforementioned clustering technique, in which the parameters describing the stiffness of the respective soft tissue are sorted according to a predefined number of intervals. By doing so, the number of unknowns associated with the elastography inverse problem can be chosen explicitly. A positive side effect of this method is the reduction of artificial noise in the data (smoothing of the solution). The performance and the rate of convergence of the resulting numerical formulation are critically analyzed by numerical examples. © 2010 Institute of Physics and Engineering in Medicine.view abstract 10.1088/0031-9155/55/7/016 **Low cycle lifetime assessment of Al2024 alloy**

Khan, S. and Vyshnevskyy, A. and Mosler, J.*International Journal of Fatigue*32 (2010)The 2024-T351 aluminum alloy is extensively used for fabricating aircraft parts. This alloy shows a relatively low ductility at room temperature and is generally heat treated in various conditions to suit particular applications. The present study experimentally and numerically analyzes the damage mechanism of an Al2024-T351 plate (short transverse direction) subjected to multi-axial stress states. The purpose of this work is to predict the cyclic lifetime of the considered alloy, based on the local approach of damage evolution using continuum damage modeling (CDM). The experimental program involves different kinds of specimens and loading conditions. Monotonic and cyclic tests have been conducted in order to measure the mechanical response and also to perform micromechanical characterization of damage and fracture processes. The cyclic plasticity behavior has been characterized by means of smooth cylindrical specimens. For analyzing the evolution of plastic deformation and damage under multi-axial stress conditions, cyclic loading tests in the low cycle regime have been conducted on different round notched bars. The predictions of the CDM were compared to the experimentally observed mechanical response and to the micromechanical characterization of damage. Emphasis was placed on the prediction of the number of cycles to failure. © 2010 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijfatigue.2010.01.014 **On the implementation of rate-independent standard dissipative solids at finite strain - Variational constitutive updates**

Mosler, J. and Bruhns, O.T.*Computer Methods in Applied Mechanics and Engineering*199 (2010)This paper is concerned with an efficient, variationally consistent, implementation for rate-independent dissipative solids at finite strain. More precisely, focus is on finite strain plasticity theory based on a multiplicative decomposition of the deformation gradient. Adopting the formalism of standard dissipative solids which allows to describe constitutive models by means of only two potentials being the Helmholtz energy and the yield function (or equivalently, a dissipation functional), finite strain plasticity is recast into an equivalent minimization problem. In contrast to previous models, the presented framework covers isotropic and kinematic hardening as well as isotropic and anisotropic elasticity and yield functions. Based on this approach a novel numerical implementation representing the main contribution of the paper is given. In sharp contrast to by now classical approaches such as the return-mapping algorithm and analogously to the theoretical part, the numerical formulation is variationally consistent, i.e., all unknown variables follow naturally from minimizing the energy of the considered system. Consequently, several different numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Extending previously published works on variational constitutive updates, the advocated model does not rely on any material symmetry and therefore, it can be applied to a broad range of different plasticity theories. As two examples, an anisotropic Hill-type and a Barlat-type model are implemented. Numerical examples demonstrate the applicability and the performance of the proposed implementation. © 2009 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cma.2009.07.006 **On the Implementation of Variational Constitutive Updates at Finite Strains**

Mosler, J. and Bruhns, O. T.*Iutam Symposium on Variational Concepts with Applications to the Mechanics of Materials*21 (2010)In this paper an efficient, variationally consistent, algorithmic formulation for rate-independent dissipative solids at finite strain is presented. Focusing on finite strain plasticity theory and adopting the formalism of standard dissipative solids, the considered class of constitutive models can be defined by means of only two potentials being the Helmholtz energy and the yield function (or equivalently, a dissipation functional). More importantly, by assuming associative evolution equations, these potentials allow to recast finite strain plasticity into an equivalent, variationally consistent minimization problem, cf. [1-4]. Based on this physically sound theoretical approach, a novel numerical implementation is discussed. Analogously to the theoretical part, it is variationally consistent, i.e., all unknown variables follow naturally from minimizing the energy of the respective system. Extending previously published works on such methods, the advocated numerical scheme does not rely on any material symmetry regarding the elastic and the plastic response and covers isotropic, kinematic and combined hardening, cf. [5, 6].view abstract 10.1007/978-90-481-9195-6_15 **Variationally consistent modeling of finite strain plasticity theory with non-linear kinematic hardening**

Mosler, J.*Computer Methods in Applied Mechanics and Engineering*199 (2010)Variational constitutive updates provide a physically and mathematically sound framework for the numerical implementation of material models. In contrast to conventional schemes such as the return-mapping algorithm, they are directly and naturally based on the underlying variational principle. Hence, the resulting numerical scheme inherits all properties of that principle. In the present paper, focus is on a certain class of those variational methods which relies on energy minimization. Consequently, the algorithmic formulation is governed by energy minimization as well. Accordingly, standard optimization algorithms can be applied to solve the numerical problem. A further advantage compared to conventional approaches is the existence of a natural distance (semi metric) induced by the minimization principle. Such a distance is the foundation for error estimation and as a result, for adaptive finite elements methods. Though variational constitutive updates are relatively well developed for so-called standard dissipative solids, i.e., solids characterized by the normality rule, the more general case, i.e., generalized standard materials, is far from being understood. More precisely, (Int. J. Sol. Struct. 2009, 46:1676-1684) represents the first step towards this goal. In the present paper, a variational constitutive update suitable for a class of nonlinear kinematic hardening models at finite strains is presented. Two different prototypes of Armstrong-Frederick-type are re-formulated into the aforementioned variationally consistent framework. Numerical tests demonstrate the consistency of the resulting implementation. © 2010 Elsevier B.V.view abstract 10.1016/j.cma.2010.03.025

#### computational mechanics

#### deformation

#### fatigue

#### finite element method

#### mechanical properties

#### numerical methods

#### thermodynamics