Universität Stuttgart

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Open Access
Approaches to the description of anisotropic material behaviour at finite elastic and plastic deformations : theory and numerics
(2004) Apel, Nikolas; Miehe, Christian (Prof. Dr.-Ing.)
The present work deals with purely macroscopic descriptions of anisotropic material behaviour. Key aspects are new developments in the theory and numerics of anisotropic plasticity. After a short discussion of the classification of solids by symmetry transformations a survey about representation theory of isotropic tensor functions and tensor polynomials is given. Next alternative macroscopic approaches to finite plasticity are discussed. When considering a multiplicative decomposition of the deformation gradient into an elastic part and a plastic part, a nine dimensional flow rule is obtained that allows the modeling of plastic rotation. An alternative approach bases on the introduction of a metric-like internal variable, the so-called plastic metric, that accounts for the plastic deformation of the material. In this context, a new class of constitutive models is obtained for the choice of logarithmic strains and an additive decomposition of the total strain measure into elastic and plastic parts. The attractiveness of this class of models is due to their modular structure as well as the affinity of the constitutive model and the algorithms inside the logarithmic strain space to models from geometric linear theory. On the numerical side, implicit and explicit integration algorithms and stress update algorithms for anisotropic plasticity are developed. Their numerical efficiency crucially bases on their careful construction. Special focus is put on algorithms that are suitable for variational formulations. Due to their (incremental) potential property, the corresponding algorithms can be formulated in terms of symmetric quantities. A reduced storage effort and less required solver capacity are key advantages compared to their standard counterparts.
Open Access
Aspects of energy minimization in solid mechanics : evolution of inelastic microstructures and crack propagation
(2007) Gürses, Ercan; Miehe, Christian (Prof. Dr.-Ing.)
This work deals with theoretical energy minimization principles and the development of associated computational tools for the description of microstructure evolution and fracture in solid mechanics. The thesis consists of two parts: (i) The description of inelastic deformation microstructures and their evolution in non-convex unstable solids and (ii) the development of a variational framework for configurational-force-driven brittle fracture based on energy minimization principles. In the first part, a general framework is developed for the treatment of material instabilities and microstructure developments in inelastic solids. Material instabilities and microstructure developments are interpreted as the outcome of non(quasi)-convex variational problems which often suffer from the lack of solutions in the classical sense. The proposed framework is based on a mathematical relaxation theory which is associated with the replacement of non-quasiconvex potentials with their generalized convex envelopes. Furthermore, deformation microstructures and their evolution are studied for three different constitutive material responses: the symmetry-breaking martensitic phase transformations, the single-slip crystal plasticity and the isotropic damage mechanics. For this purpose specific numerical relaxation algorithms are proposed for each constitutive response. The performance of numerical relaxation schemes is presented by several representative examples. In the second part, a variational formulation of quasistatic brittle fracture in elastic solids is outlined and a finite-element-based computational framework is proposed for the two- and three-dimensional crack propagation. The starting point is a variational setting that recasts a monotonic quasistatic fracture process into a sequence of incremental energy minimization problems. The proposed numerical implementation exploits this variational structure. It introduces discretized crack patterns with configurational-force-driven incremental crack segment and crack surface releases. These releases of crack segments and surfaces constitute a sequence of positive definite subproblems with successively decreasing overall stiffness, providing an extremely robust algorithmic setting in the postcritical range. The formulation is embedded into an accompanying r-adaptive crack-pattern adjustment procedure with configurational-force-based indicators in conjunction with crack front constraints. The performance of the proposed algorithm is demonstrated by means of several two- and three-dimensional crack propagation examples and comparisons with experiments.
Open Access
Computational modeling of ferromagnetics and magnetorheological elastomers
(2014) Ethiraj, Gautam; Miehe, Christian (Prof. Dr.-Ing.)
The aim of this work is to present new variational-based computational modeling approaches for selected materials that have coupled magnetic and mechanical properties. In order to solve the magnetomechanically coupled boundary value problems, we employ the finite element method and also discuss certain aspects that are peculiar to magnetomechanical problems such as the unity constraint on the magnetization, the incorporation of the surrounding free space, the micromechanics of the coupled response etc. Thus, we present continuum models motivated by underlying physical phenomena at the micro- and nano-scale that are embedded in appropriate variational-based finite element frameworks allowing the simulation and visualisation of finite-sized bodies. Specifically, we present the computational modeling of two types of magnetostrictive materials namely, 1. Ferromagnetic materials with magnetic domain microstructures that evolve dissipatively. Our approach is based on Brown's theory of micromagnetics and now extended and implemented within the computationally powerful finite element method in which the focus is on the geometric consistency of the numerical setting. 2. Magnetorheological Elastomers (MREs) where we will introduce a modular approach for the construction of micromechanically motivated models for the constitutive response of such materials. Motivated by Toupins work on the elastic dielectric, we will discuss the variational principle and the implementation in the finite element framework. In modeling the above materials, we span the range of scales involved in continuum magnetomechanics, and also highlight the variety of challenges that exist in the field. From the theoretical and computational standpoint, this work aims to contribute to the ultimate goal of construction of a compatible hierarchy of models for magnetomechanically coupled materials.
Open Access
Hybrid micro-macro modeling of texture evolution in polycrystal plasticity based on microstructural reorientation continua
(2013) Zimmermann, Ilona Andrea; Miehe, Christian (Prof. Dr.-Ing.)
The present work deals with the modeling of evolving crystal orientation microstructures in finite polycrystal plasticity and its impact on the macroscopic material behavior by means of a two-scale approach. A micro-mechanical plasticity model is developed that locally accounts for microscopic structural changes in the form of grain reorientations. The algorithmic treatment captures in a numerically efficient manner the crystal reorientation for evolving face- and body-centered cubic textures. Thereby, the parametrization of rotations is carried out in the Rodigues space. The performance is demonstrated by means of representative numerical examples. As a key ingredient the crystallographic texture is responsible for the development of macroscopic anisotropy, entailing the necessity of a multiscale approach for appropriately predicting the material behavior. Crystal orientation distribution functions govern the evolution of structural tensors, representing in a homogenized sense the crystal reorientation within a model-inherent scale bridging technique. The texture estimation is incorporated in a modular format into a micro-macro model resulting in a computationally manageable approach compared to straightforward homogenization-based multiscale methods, such as e.g. FE2. A macro-mechanical model of anisotropic finite plasticity is based on evolving structural tensors accounting for the texture-induced macroscopic anisotropy. The general framework for the micro-macro modeling is a purely phenomenological setting of anisotropic plasticity in the logarithmic strain space. The capabilities and computationally efficiency of his hybrid two-scale model of finite polycrystalline plasticity is demonstrated by means of a variety of numerical examples including the comparison with benchmark analyses and experimental observations.
Open Access
Incompatibility and instability based size effects in crystals and composites at finite elastoplastic strains
(2006) Becker, Martin; Miehe, Christian (Prof. Dr.-Ing.)
The purpose of this work is the description of length scale dependencies in nonhomogeneously deforming crystals and the elimination of size dependencies in classical homogenization approaches for instable elastoplastically deforming composites. Key aspects, on the side of the investigation of size effects in crystals, are a comprehensive discussion of the underlying micromechanical interpretation, the deformation geometry and related experimental observations. These include a detailed incompatibility analysis and an extensive discussion of the dislocation density tensor and the storage of geometrically necessary dislocations. A dislocation density based strain gradient crystal plasticity model is developed as a main outcome of these investigations. This model is subsequently treated in the context of a mixed finite element formulation or alternatively in a more efficient manner through an extended standard local formulation. The developments are validated through various numerical examples which cover also a comparison with experimental observations or predictions obtained through alternative approaches such as discrete dislocation simulations. In view of an investigation of size dependencies in the homogenization analysis of instable inelastic composites, first criteria for an instability analysis on the micro- as well as the macro-scale are developed. The underlying basis is an incremental variational formulation of the homogenization problem. This allows for an investigation of the interaction between micro- and macro-instabilities and the development of a non-convex homogenization approach for inelastic composites at finite strains. The implications of the non-convex homogenization approach which, as a key point additionally determines the relevant microstructure size, are finally demonstrated for several examples.
Open Access
Material forces in finite inelasticity and structural dynamics : topology optimization, mesh refinement and fracture
(2008) Zimmermann, Dominik; Miehe, Christian (Prof. Dr.-Ing.)
The present work serves two major purposes. On the one hand, theoretical approaches to configurational mechanics are elaborated. For inelastic problems, the spatial and material equilibrium conditions are derived by means of a global dissipation analysis. In the dynamical framework, a variational formulation based on Hamilton's principle is established inducing the balances of physical momentum, material pseudomomentum and kinetic energy. On the other hand, configurational-force-based computational algorithms are developed. At first, configurational forces are exploited in the context of topology optimization. The theoretical basis is provided by a dual variational formulation of finite elastostatics. This scenario is applied to the r-adaptive optimization of finite element meshes and the optimization of truss structures. In the second step, a configurational-force-based strategy for h-adaptvity is presented. The discrete version of the material balance equation is exploited to formulate global and local refinement criteria controlling the overall decision on mesh refinement and the local refinement procedure. The method is specified for problems of finite elasticity and plasticity including thermal and dynamical effects as well. Finally, a configurational-force-driven procedure for the simulation of crack propagation in brittle materials is introduced. The algorithm bases on the separation of the geometry model and the finite element mesh. The process of crack propagation is carried out by a structural update of the underlying geometry model. The generation of the new triangulation incorporates a configurational-force-based adaptive refinement criterion. The capabilities of the derived algorithms are demonstrated by means of a variety of numerical examples including the comparison with benchmark analyses and experimental observations.
Open Access
Mechanics of nonlocal dissipative solids: gradient plasticity and phase field modeling of ductile fracture
(Stuttgart : Institut für Mechanik (Bauwesen), Lehrstuhl I, Universität Stuttgart, 2016) Aldakheel, Fadi; Miehe, Christian (Prof. Dr.-Ing.)
The underlying work is concerned with the development of physically-motivated constitutive models for the description of size effects within the context of inelastic deformations. A key aspect of this thesis is to develop a theoretical and computational framework for gradient-extended dissipative solids. It incorporates spatial gradients of selected micro-structural fields that account for length scale effects and describe the evolving dissipative mechanisms. In contrast to classical theories of local continuum mechanics, where the internal variables are determined by ordinary differential equations (ODEs), these global micro-structural (order parameter) fields are governed by partial differential equations (PDEs) and boundary conditions reflecting the continuity of these variables. The proposed framework for gradient-extended dissipative solids is first used to address the development of phenomenological theories of strain gradient plasticity. The corresponding model guarantees from the computational side a mesh-objective response in the post-critical ranges of softening materials. In this regard, a mixed variational principle for the evolution problem of gradient plasticity undergoing small and large strains is developed. A novel finite element formulation of the coupled problem incorporating a long-range hardening/softening parameter and its dual driving force is also proposed. A second employment of the introduced framework is related to the thermo-mechanical coupling in gradient plasticity theory within small strain deformations. Two global solution procedures for the thermo-mechanically coupled problem are introduced, namely the product formula algorithm and the coupled-simultaneous solution algorithm. For this purpose, a family of mixed finite element formulations is derived to account for the coupled thermo-mechanical boundary-value problem. A further application of the proposed framework deals with the phase-field modeling of ductile fracture undergoing large strains. To this end, a novel variational-based framework for the phase-field modeling of ductile fracture in gradient-extended elastic-plastic solids is proposed. Herein, two independent length scales, that regularize both the plastic response as well as the crack discontinuities, are introduced. This ensures that the failure zone of ductile fracture takes place inside the plastic zone, and guarantees from the computational perspective mesh objectivity in the post-critical range. The performance of these models is tested on a broad range of homogeneous and heterogeneous representative numerical simulations.
Open Access
Mehrskalenmodelle in der Festkörpermechanik und Kopplung von Mehrgittermethoden mit Homogenisierungsverfahren
(2005) Bayreuther, Claus; Miehe, Christian (Prof. Dr.-Ing.)
Ziel dieser Arbeit ist die Formulierung von nichtlinearen homogenisierten Ersatzmodellen für mikroheterogene Materialien und die Konstruktion problemabhängiger Transferoperatoren für Mehrgitterverfahren. Beide Themenkomplexe haben eine effiziente Beschreibung heterogener Festkörperstrukturen zum Ziel. Die Simulation von Verbundstrukturen stellt ein sehr komplexes Problem dar, insbesondere wenn die Skalenabhängigkeit des Werkstoffs miteinbezogen wird. Dies gründet darin, daß die Dimension des makroskopischen Randwertproblems und die Abmessungen der Heterogenitäten auf der Mikroskale stark voneinander abweichen können. In diesem Fall ist eine effiziente Modellierung nur durch geeignete Mehrskalenbildung möglich. In dieser Arbeit werden analytische und neu entwickelte numerische Homogenisierungsmodelle für Skalenübergänge aufbereitet. Im Gegensatz zu analytischen Konzepten gestattet die Methode der Finite Elemente universelle Einsatzmöglichkeiten der numerischen Modelle. Die neuen numerischen Ansätze basieren auf diskreten Variationsprinzipien, deren Umsetzung auf der Mikroskale die Lösung eines Randwertproblems mit speziellen Randbedingungen erfordert: (i) lineare oder (ii) periodische Randverschiebungen oder (iii) homogene Randspannungen auf dem Rand einer charakteristischen Mikrostruktur. Die Effizienz der neuen numerischen Ersatzmodelle wird anhand repräsentativer Einheitszellen verifiziert. Bei kleiner Skalenseparation führt die direkte numerische Diskretisierung des Verbundwerkstoffs in der Regel auf großdimensionierte Gleichungssysteme, die den Einsatz schneller Löser, wie Mehrgitterverfahren, bedingen. Bei Mehrgittermethoden liegt die Schwierigkeit in der Konstruktion geeigneter Transferoperatoren. In dieser Arbeit wird dieses Problem durch Einbeziehung der neu entwickelten Homogenisierungstechniken gelöst. Effizienz und Anwendungsgrenzen der neuen Transferoperatoren werden an typischen Modellproblemen im Vergleich zu alternativen Konzepten aufgezeigt.
Open Access
Micro-macro approaches to rubbery and glassy polymers : predictive micromechanically-based models and simulations
(2007) Göktepe, Serdar; Miehe, Christian (Prof. Dr.-Ing. habil.)
This work is concerned with the development of physically motivated constitutive models for the description of the material behavior of rubbery and glassy polymers. The particular focus of the thesis is placed on elasticity, finite viscoelasticity, deformation-induced Mullins-type damage in rubbery polymers, and finite viscoplasticity of amorphous glassy polymers. The models developed possess the intrinsic character of a micro-macro transition that, in turn, allows us to incorporate the physical mechanisms stemming from a micro-structure of the material through geometrically well defined kinematic measures and in terms of physically motivated material parameters. The proposed approaches make use of a micro-structure that is symbolized by a unit sphere, the so-called micro-sphere. The surface of the micro-sphere represents a continuous distribution of chain orientations in space. A key idea of the proposed constitutive framework may be considered as a two-step procedure that incorporates the set up of micromechanically-based constitutive models for a single chain orientation and the definition of the macroscopic stress response through a directly evaluated homogenization of state variables. The disribution of micro-state variables are defined on the micro-sphere of space orientations in a discrete manner. The proposed models are further furnished with the associated algorithmic procedures that perform the update of internal variables and computation of stresses and tangent moduli in a way consistent with the employed integration scheme. The modeling performance of the models is tested against broad range of homogeneous and inhomogeneous experimental data with particular regard to their predictive simulation capabilities.
Open Access
Multi-level descriptions of failure phenomena with the strong discontinuity approach
(2014) Raina, Arun; Miehe, Christian (Prof. Dr.-Ing.)
The ever increasing demand of advanced engineered products also pushes the strengths of the materials used to their theoretical limits. It becomes crucially important to understand the behavior of such materials during failure for an efficient and safe design of the product. This thesis aims at the physical-based numerical modeling of complex failure phenomena in engineering materials, categorized into hard matter and soft matter. In Part I of this thesis, a modification of the well established strong discontinuity approach to model failure phenomena in hard matter by extending it to multiple levels is proposed. This is achieved by the resolution of the overall problem into a main boundary value problem and identified sub-domains based on the concepts of domain decomposition. Those sub- domains are subsequently adaptively discretized during run-time and comprise the so- called sub-boundary value problem to be solved simultaneously with the main boundary value problem. To model failure, only the sub-elements of those sub-boundary value problems are treated by the strong discontinuity approach which, depending on their state of stress, may develop cracks and shear bands. A single finite element of the main boundary value problem can therefore simulate the propagation of multiple propagating strong discontinuities specially arising for simulations of crack branching. The solutions of the different sub-boundary value problems are transferred to the main boundary value problem based on concepts of domain decomposition. The applied boundary conditions are also modified to account for the possible multiple jumps in the displacement fields. It is shown through the simulation of solids undergoing dynamic fracture that the modification allows to predict the onset of crack branching without the need for any artificial crack branching criterion. A close agreement with experiments of the simulation results in terms of micro- and macro branching in addition to studying certain key parameters like critical velocity, dynamic stress intensity factor, and the strain energy release rate at branching is found. In Part II of this thesis, failure phenomena in soft matter is modeled for which an advanced homogenization approach to model the highly anisotropic and non-linear stiffening response at finite strains is developed first. The constituent one-dimensional elements are modeled as linear elastic, by experimental justification, which are modified in the lower strain regime to account for the inherent fiber undulations and the associated fiber unfolding phenomena. Reorientation of these fibers is identified as one primary mechanism for the overall macroscopic stiffening which is achieved by a new bijective mapping asymptotically aligning these fibers with the maximum loading direction in the referential orientation space. A rate-independent evolution law for this map is sought by a physically motivated assumption to maintain the overall elastic framework of the proposed formulation. A closed form solution to the new evolution law is also presented which allows faster computation of updating orientations without resorting to numerical integration or storing history variables. The unit vectors upon reorientation in the referential orientation space are then mapped to the spatial orientation space by the macro deformation gradient to compute the macroscopic Kirchhoff stress and the associated spatial elasticity modulus. A direct comparison of the numerical results with the experimental results from the literature is made which demonstrates the predictive capabilities of the proposed formulation. Finally, the finite deformation extended strong discontinuity approach is utilized to simulate boundary value problems of failure in nonwoven felts. The simulation results of failure show a satisfactory agreement with the experimental data from literature.
Open Access
On the computational modeling of micromechanical phenomena in solid materials
(2013) Linder, Christian; Miehe, Christian (Prof. Dr.-Ing. habil.)
This work aims to contribute to the research on the constitutive modeling of solid materials, by investigating three particular micromechanical phenomena on three different length scales. The first microscopic phenomenon to be considered on the macroscopic scale is the process of failure in solid materials. Its characteristic non-smoothness in the displacement field results in the need for sophisticated numerical techniques in case one aims to capture those failure zones in a discrete way. One of the few finite element based methods successfully applied to such challenging problems is the so called strong discontinuity approach, for which failure can be described within the individual finite elements. To avoid stress locking, a higher order approximation of the resulting strong discontinuities is developed in the first part of this work for both, purely mechanical as well as electromechanical coupled materials. A sophisticated crack propagation concept relying on a combination of the widely used global tracking algorithm and the computer graphics based marching cubes algorithm is employed to obtain realistic crack paths in three dimensional simulations. Secondly, materials with an inherent network microstructures such as elastomers, hydrogels, non-woven fabrics or biological tissues are considered. The development of advanced homogenization principles accounting for such microstructures is the main focus in the second part of this work to better understand the mechanical and time-dependent effects displayed by such soft materials. Finally, the incorporation of wave functions into finite element based electronic structure calculations at the microscopic scale aims to account for the fact that the properties of condensed matter as for example electric conductivity, magnetism as well as the mechanical response upon external excitations are determined by the electronic structure of a material.
Open Access
On the formulation and numerical implementation of dissipative electro-mechanics at large strains
(2010) Rosato, Daniele; Miehe, Christian (Prof. Dr.-Ing. habil.)
In recent years an increasing interest in functional materials such as erroelectric polymers and ceramics has been shown. For those materials, viscous effects or electric polarizations cause hysteresis phenomena accompanied with possibly large remanent strains and rotations. In this work aspects of the formulation and numerical implementation of dissipative electro-mechanics at large strains are outlined. In particular continuous and discrete variational formulations for the treatment of the non-linear dissipative response of electro-mechanical solids are developed and these formulations are adapted to the modeling of the hysteretic material response of piezoceramics and ferroelectric polymers under electrical loading. The point of departure is a general internal variable formulation that determines the hysteretic response of the material as a generalized standard medium in terms of an energy storage and a rate-dependent dissipation function. Consistent with this type of standard dissipative continua, an incremental variational formulation of the coupled electro-mechanical boundary-value-problem is developed. The variational formulation for a setting based on a smooth rate-dependent dissipation function which governs the hysteretic response is specified. Further, the geometric nature of dissipative electro-mechanics is underlined. An important aspect is the numerical implementation of the coupled problem. The discretization of the two-field problem appears, as a consequence of the proposed incremental variational principle, in a symmetric and very compact format. Further, constitutive assumptions which account for specific problems arising in the geometric nonlinear setting are discussed. With regard to the choice of the internal variables entering the constitutive functions, a critical point are the kinematic assumptions. Here, the multiplicative decomposition of the local deformation gradient into reversible and remanent parts as well as the introduction of a remanent metric are discussed. Such a formulation allows us to reproduce the dielectric and butterfly hysteresis responses characteristic of the ferroelectric materials together with their rate-dependency and to account for macroscopically non-uniform distribution of the polarization in the specimen together with large attained deformations. The performance of the proposed methods is demonstrated by means of a spectrum of benchmark problems which eventually show large deformations.
Open Access
Phase field modeling of fracture in rubbery and glassy polymers at finite thermo-viscoelastic deformations
(2015) Schänzel, Lisa-Marie; Miehe, Christian (Prof. Dr.-Ing.)
The goal of this work is to provide a theoretical and computer based model for brittle and ductile fracture mechanics, which enables the modeling of complex fracture phenomena with large deformations. A central aspect of this work is to provide a comprehensive theoretical study of a phase field model of fracture and its application towards the modeling of crack initiation and growth in rubbery and glassy polymers at finite thermo-viscoelastic deformations. The other main aspects are the development of new algorithms for crack propagation and investigations towards the predictive quality of these new methods. Fracture is the partial or full separation of an object or material into two or more pieces under the influence of stress. In 1920, Griffith introduced the so-called energy release rate for brittle elastic materials which is the energy required for crack propagation and thus, created the energetic fracture criterion. Fracture mechanic models exist for the description of both sharp and diffusive crack discontinuities. Models describing sharp crack discontinuities include cohesive-zone models or configurational-force-driven models. These, however, suffer in situations with complex crack evolution. In contrast, phase field type diffusive crack approaches are smooth continuum formulations. These avoid the modeling of discontinuities and thus allow a straightforward computation of complex curved crack and fracture phenomena such as crack initiation, crack branching or crack arrest. This work presents the application of a fracture phase field model towards the modeling of crack initiation and growth in rubbery and glassy polymers at finite thermo-viscoelastic deformations. Due to their molecular structure, polymeric materials show a wide range of both, mechanical material behavior as well as fracture behavior. Rubbery polymers show highly nonlinear elasticity, characterized by the typical S-shaped uniaxial nominal stress-stretch relation. The origin of the characteristic of rubber to undergo large elastic deformations, is the ability of the coiled and entangled polymer chains to elongate and disentangle under tensile stress, such that the macromolecules align in the direction of the applied force. The complex three-dimensional structure of rubbery polymers consists of chemically crosslinked and entangled macromolecules. These take up compact, random configurations, which lead to complex viscoelastic material behavior. Thermoelastic polymers and elastomers show rate dependent material behavior. During slow deformation, the molecular segments can easily rotate and realign. Thus, the entanglements contribute little to the stiffness and the material deforms almost completely elastically. However, with increasing deformation rate, the transformations of the molecular segments can no longer keep up with the rate of defomation. Hence, the stiffness of the material increases. Macroscopic fracture of rubbery polymers is a result of the failure of the molecular network. Crack propagation takes place when molecular chains whose crosslinks lie on opposite sides of the crack plane are broken. These considerations resulted in the definition of a micromechanically motivated energy release rate, necessary for crack propagation. Crack growth in rubbery polymers is rarely brittle, but mostly a gradual tearing of the material under constant energy consumption, which strongly depends on the velocity of the crack tip. Reasons for this are dissipative effects in both the bulk material and the process zone. An increase in temperature results in an increasing polymer molecule mobility, a decrease of dissipative effects and thus a decrease of energy release rate. Amorphous glassy polymers exhibit a small range of linear elastic deformation. The strength is limited by brittle fracture, cold drawing, shear yielding or crazing. Cold drawing is characterized by a small range of linear elastic strains up to the yield point, followed by a large amount of plastic deformation during which, the stress remains almost constant. Macroscopically a stable neck spreads over the object in question within which the molecules are stretched and align in the direction of the applied force. Crazing is also termed dilatational normal stress yielding and is a plastic deformation mechanism. Crazes consist of dense arrays of fibrils separated by voids. They can grow in width and length until the fibrils break down, which eventually leads to structural failure. Shear yielding and crazing are not completely independent, nor do they exclude each other. Changing the temperature or the rate of deformation and thus modifying the mobility and reaction of the molecules, causes a change of yield stress and brittle fracture stress. Thus a transition from ductile to brittle material response can take place.
Open Access
Static and dynamic homogenization analyses of discrete granular and atomistic structures on different time and length scales
(2006) Dettmar, Joachim Peter; Miehe, Christian (Prof. Dr.-Ing.)
This work deals with scale bridging methods for discrete microscopic granular and nanoscopic atomistic aggregates of particles between different length and time scales. The bridging between the scales is achieved by direct homogenization of micro- and nanoscopic physical quantities. In the first part of the work, static homogenization techniques for granular materials are developed where a distinct definition of a granular microstructure serves as the starting point for the development of homogenization definitions. Three new boundary constraints are consistently derived from classical continuous definitions and implemented in a strain-driven environment. The finite-sized character of the particles is accounted for in the formulation. With regard to stiffness, it is shown that the periodic surface constraints are bounded by the linear deformation- and uniform traction constraints. Additionally, true dual-scale analyses of granular structures at large strains are performed. The continuum approach on the coarse scale employs the finite element method which serves as a numerical tool without any constitutive assumptions as the physical input is solely governed by the granular microstructures. The second part of the work covers dynamic homogenization techniques in connection with the classical molecular dynamics method for atomistic simulations. A uniform traction constraint is developed and it is shown that this formulation is the only suitable choice as it allows for a computational modeling of defects and cracks in a nanosystem. The constraint is implemented in a deformation-controlled environment allowing for computational treatments in coarse scale continuum methods. The dynamic homogenization incorporates the kinetics of all atoms in the aggregate. Several numerical examples in all sections of the thesis round off the discussion of the static and dynamic homogenization techniques for discrete structures.
Open Access
Temperaturabhängige Beschreibung visko-elasto-plastischer Deformationen kurzglasfaserverstärkter Thermoplaste : Modellbildung, Numerik und Experimente
(2004) Rieger, Sonja; Miehe, Christian (Prof. Dr.-Ing.)
Im Rahmen dieser Arbeit wird ein visko-elasto-plastisches Materialmodell zur Beschreibung des realen Materialverhaltens eines kurzglasfaserverstärkten Thermoplasts entwickelt. Hierzu müssen zunächst zahlreiche Experimente durchgeführt werden um das phänomenologische Verhalten des kurzfaserverstärkten Thermoplastes Polyamid 66 mit 35 % Glasfaseranteil charakterisieren zu können. Untersucht wird sowohl das Verhalten im Kurzzeitbereich, wie auch das Langzeitverhalten des faserverstärkten Thermoplastes unter verschiedenen Temperaturen. Um die anisotropen Effekte des Materials beschreiben zu können, werden die Experimente an Proben mit unterschiedlichen Faserorientierungen durchgeführt. Für die beobachteten Phänomene wird mit Hilfe logarithmischer Verzerrungen ein physikalisch und geometrisch nichtlineares Materialmodell entwickelt, dass sich aus einem elastoplastischen Teilwerkstoffgesetz mit Schädigung und kinematischer Verfestigung und mehreren viskoelastischen Teilwerkstoffgesetzen zusammensetzt. Zusätzlich wird der Temperatureinfluss für das viskoelastische Teilmodell unter Verwendung des Zeit-Temperatur-Verschiebungsprinzips berücksichtigt. Hierbei werden die Relaxationszeiten durch einen Temperaturverschiebungsfaktor bei einer Temperaturerhöhung verkleinert. Als weiterer thermischer Effekt wird die anisotrope thermische Wärmeausdehnung im Modell berücksichtigt. Nachdem das mathematische Modell alle wesentlichen Eigenschaften des Materials qualitativ wiedergeben kann, werden die Materialparameter mit Hilfe nichtlinearer Optimierung bestimmt um eine quatitative Übereinstimmung von Simulation und Experiment zu erlangen. Abschließend wird das mathematische Modell in das kommerzielle Finite Element Programm ABAQUS eingebunden und anhand eines repräsentativen mehrdimensionalen Beispiels validiert.
Open Access
Thermoviscoplasticity of glassy polymers : experimental characterization, parameter identification and model validation
(2010) Méndez Diez, Joel; Miehe, Christian (Prof. Dr.)
This work is concerned with the experimental characterization of the mechanical response of glassy polymers under various deformation modes, strain rates and temperatures together with the application of a framework of thermo-elasto-visco-plasticity in the logarithmic strain space recently developed in our group. The presented experiments are mainly classified based on the type of deformation in homogeneous and inhomogeneous and based on the applied strain rate in isothermal and thermomechanical. Homogeneous tests are understood to be uniaxial compression and plane strain compression experiments where the temperature does not rise due to deformation. Inhomogeneous experiments performed under tension are carried out together with unconventional equipment that allows a deeper insight into the necking phenomenon in glassy polymers. Discussions on the preprocessing, acquisition and post-processing of the experimental data are included. In the computational part of this work the plastic flow is approximated by means of two different micromechanically motivated constitutive theories for glassy polymers. The identification of the necessary material parameters and the simulations obtained therefrom give the foundation for a critical review of the capabilities of the employed constitutive laws. Additionally, thermomechanical experiments are simulated by one of the presented constitutive models to evaluate the capacity of the model at higher strain rates.
Open Access
A variational framework for gradient-extended dissipative continua : application to damage mechanics, fracture, and plasticity
(2011) Welschinger, Fabian Richard; Miehe, Christian (Prof. Dr.-Ing.)
The thesis addresses the development of a variational-based framework for gradient-type standard dissipative solids. A focus lies on the design of theoretical and computational approaches towards the description of length-scale effects in inelastically deforming solids. A strong emphasis is put on a unifying theoretical and numerical treatment of the incremental variational formulation that is applied to a broad class of gradient-type solids with intrinsic length scales. The coupled, symmetric multi-field formulation is first used to model gradient-type damage mechanics that overcomes drawbacks of local constitutive damage models regarding mesh sensitivity. A second application of the variational-based framework for gradient-type solids is concerned with the phase field modeling of fracture, allowing for the prediction of curvilinear crack patterns, crack kinking, and crack initiation in solids free of imperfections. This formulation avoids the modeling of sharp discontinuities usually done in classical approaches towards fracture and turns out to be conceptually in line with the previously discussed model of gradient-type damage mechanics. A challenge of the phase field modeling of fracture arises with regard to the approximate description of the crack topology. Accurate results demand the employment of highly densified finite element meshes in the crack evolution zone. An improvement of the numerical efficiency is obtained by an h-adaptive solution procedure that is exclusively governed by discrete configurational forces. A last application of the proposed framework covers models of phenomenological plasticity with gradient-type hardening at small and large deformations. These models allow for the regularization of shear bands and the description of the so-called Hall-Petch effect.
Open Access
Variational homogenization in electro-mechanics : from micro-electro-elasticity to electroactive polymers
(2014) Zäh, Dominic; Miehe, Christian (Prof. Dr.-Ing.)
In recent years an increasing interest in functional or smart materials such as ferroelectric polymers and ceramics has been shown. Regarding the technical implementation of smart systems a broad variety of physically-based phenomena and materials are available, where some of the most important coupling effects are the shape memory effect, magnetostriction, electrostriction, and piezoelectricity. Typical fields of application are adaptive or controlled systems such as actuators and sensors, micro-electro-mechanical systems (MEMS), fuel injectors for common rail diesel engines, ferroelectric random access memories, and artificial muscles used in robotics. A highly interesting class of these materials are piezoceramics, coming up with short response times, high precision positioning, relatively low power requirements, and high generative forces, providing an excellent opportunity for mass production. Typical examples of such materials are barium titanate and lead zirconate titanate crystals and polycrystals, which exhibit linear and nonlinear coupling phenomena as well as hysteresis under high cyclic loading. At the microscale level, these materials are composed of several homogeneously polarized regions, called ferroelectric domains, whose evolution in time is driven by external electric fields and stresses applied to a sample of the material. Ferroelectric domains are regions of parallel and hence aligned polarization. Electric poling can be achieved by the application of a sufficiently strong electric field, inducing the reorientation and alignment of spontaneous polarization. As a consequence, piezoceramics exhibit a macroscopic remanent polarization. On the other hand, there are electroactive polymers (EAPs) responding by a (possibly large) deformation to an applied electrical stimulus, an effect discovered by the physicist Wilhem Röntgen in 1880 in an experiment on a rubber strip subjected to an electric field. They are divided into two main groups: electronic and ionic materials. The description of these effects through models of continuum physics is a subject of extensive research. Physically predictive material modeling can be performed on different length- and time scales. The classical setting of continuum mechanics develops phenomenological material models "smeared" over some continuously distributed material, where the material parameters are determined from experimental data. Nowadays developed multiscale techniques focus predominantly on the efficient bridging of neighboring length- and time scales, e.g. the incorporation of the microscopic polarization in order to predict macroscopic hysteresis phenomena. With a continuous increase in computational power and the development of efficient numerical solvers, real multiscale simulations seem to be a reachable goal. Computational homogenization schemes determine, in contrast to initially developed Voigt and Reuss bounds, the effective properties numerically. No constitutive model is explicitly assumed at the macroscale, and the material response at each point is determined by performing a separate numerical analysis at the micro-level. The macroscopic material behavior in this two-scale scenario is then determined by separate FE computations at the microscale. Main ingredients of such a framework are, on the one hand, the solution of a microscopic material model describing mechanical behavior at the representative volume element and, on the other hand, a homogenization rule determining the macroscopic stress tensor by its microscopic counterpart. Goal of these computational homogenization techniques is the modeling of the overall response based on well-defined microstructural information. Concerning the scale transition for functional materials, it is necessary to extend the homogenization principles to coupled problems, incorporating besides the mechanical displacement further primary variables such as the electric potential and the electric polarization. The key aspect of every homogenization scheme is the determination of macroscopic quantities in terms of their microscopic counterpart, driven by appropriate constraints or boundary conditions on the representative volume element. The micro-to-macro transition can be described in a canonical manner by variational principles of homogenization, determining macroscopic potentials in terms of their microscopic counterparts.
Open Access
Variational multifield modeling of the formation and evolution of laminate microstructure
(2013) Hildebrand, Felix Eberhard; Miehe, Christian (Prof. Dr.-Ing.)
The optimization of material properties and the design of new materials with tailored material behavior are among the greatest challenges in the field of computational continuum mechanics. Since the macroscopic material behavior of many technically relevant materials is very closely linked to their microstructure, a profound physical and mathematical understanding and a reliable computational prediction of the formation and evolution of this microstructure is the necessary basis for any optimization or material design. In this work, we focus on the physical and mathematical understanding and the modeling and simulation of laminate microstructure and use the modeling framework of gradient-extended standard-dissipative solids to construct a phase field model for martensitic laminate microstructure in two-variant martensitic CuAlNi and a gradient crystal plasticity model for laminate deformation microstructure in Copper with two active slip systems on the same slip plane. We derive rate- and incremental-variational as well as finite element formulations for the two models and carry out numerical simulations. Basis for our modeling are the modeling framework of gradient-extended standard-dissipative solids on the one hand, and the continuum theory of non-material sharp interfaces with interface energy on the other hand, from which we derive the condition of kinematic compatibility, jump conditions in analogy to the balance equations and the dissipation postulate for the moving interface. We consider the variational origin of the formation of laminate microstructure and identify gradient-extended modeling approaches as the suitable choice for the modeling of the formation and dissipative evolution of laminate microstructure with interface energy. Based on these considerations, we propose a phase field model for the formation and evolution of laminate microstructure in two-variant martensitic CuAlNi that is based on the variational smooth approximation of sharp topologies and contains a coherence-dependent interface energy. We show that an internal mixing approach for the bulk energy allows a clear separation of interface and bulk energy and that the model is capable of predicting the formation and dissipative evolution of martensitic laminate microstructure and size effects. Furthermore, we propose a gradient crystal plasticity model for Copper with two active slip systems on the same slip plane that allows a prediction of both the formation and evolution of plastic laminate microstructure and incorporates the effect of geometrically necessary dislocations (GNDs). The model contains a biquadratic non-convex latent hardening function and a gradient contribution based on the dislocation density tensor. The evolution equations of the plastic slips and the accumulated plastic slips are obtained by use of a rate regularization that makes use of the approximation of |x| as a*ln(cosh(x/a)) for a<<1. The model is shown to be capable of predicting the formation and evolution of deformation laminate microstructure together with length-scale effects related to GNDs.
Open Access
Zur Parameteridentifikation komplexer Materialmodelle auf der Basis realer und virtueller Testdaten
(2005) Rieger, Andreas; Miehe, Christian (Prof. Dr.-Ing.)
In der vorliegenden Arbeit werden verschiedene Methoden der Parameteridentifkation anhand von realen Meßergebnissen und virtuellen Versuchsdaten aufgezeigt. Die Parameteridentifikation führt auf eine inverse Problemstellung, die nicht exakt, aber im Sinne einer optimalen Anpassung von Simulation und Experiment, gelöst wird. Hierzu dienen die Methoden der nichtlinearen Optimierung, die eine problemabhängige Fehlerfunktion unter Beachtung von Nebenbedingungen minimiert. Zur Minimierung der Fehlerfunktion wird insbesondere die Sequentielle Quadratische Programmierung und ein Simplex Verfahren betrachtet. Für eine gradientenbasierte Optimierung werden die Sensitivitäten der in die Zielfunktion eingehenden Größen benötigt. Dies führt bei einer analytischen Bestimmung der Gradienten auf zwei entkoppelte Teile. Der eine Teil wird durch die Simulation des Experiments bestimmt. Der andere Teil beinhaltet die Spannungssensitivität, die als eine Erweiterung des algorithmischen Materialmodells betrachtet werden konnen. Die Parameteridentifikation wird auf drei unterschiedliche Klassen von Versuchen angewendet. Dies sind zum einen axiale Zug-Druckversuche, bei denen die Fehlerfunktion über den Vergleich von experimentellen und simulierten axialen Spannungen gebildet wird. Hierbei wird insbesondere auf den Vergleich von analytisch und numerisch berechneten Gradienten abgezielt. Die zweite Art von Versuchen, die in die Identifikation einbezogen werden, sind optische Verschiebungsmessungen an der Probenoberfläche. Die Fehlerfunktion wird hierbei über diskrete Knotenverschiebungen gebildet. Die Grundlage der dritten Klasse von Identifikationen bilden virtuelle Experimente. Diese Daten werden aus der Homogenisierungsanalyse heterogener Mikrostrukturen gewonnen. Die Fehlerfunktion wird über mehrdimensionale Spannungs-Verzerrungspfade aufgebaut. Alle drei Arten von Identifikationen werden anhand ausgewählter Modellprobleme dargestellt.