This thesis is concerned with a phase field model for brittle fracture.
The high potential of phase field modeling in computational fracture mechanics lies in the generality of the approach and the straightforward numerical implementation, combined with a good accuracy of the results in the sense of continuum fracture mechanics.
However, despite the convenient numerical application of phase field fracture models, a detailed understanding of the physical properties is crucial for a correct interpretation of the numerical results. Therefore, the driving mechanisms of crack propagation and nucleation in the proposed phase field fracture model are explored by a thorough numerical and analytical investigation in this work.
Elastomers and their various composites, and blends are frequently used as engineering working parts subjected to rolling friction movements. This fact already substantiates the importance of a study addressing the rolling tribological properties of elastomers and their compounds. It is worth noting that until now the research and development works on the friction and wear of rubber materials were mostly focused on abrasion and to lesser extent on sliding type of loading. As the tribological knowledge acquired with various counterparts, excluding rubbers, can hardly be adopted for those with rubbers, there is a substantial need to study the latter. Therefore, the present work was aimed at investigating the rolling friction and wear properties of different kinds of elastomers against steel under unlubricated condition. In the research the rolling friction and wear properties of various rubber materials were studied in home-made rolling ball-on-plate test configurations under dry condition. The materials inspected were ethylene/propylene/diene rubber (EPDM) without and with carbon black (EPDM_CB), hydrogenated acrylonitrile/butadiene rubber (HNBR) without and with carbon black/silica/multiwall carbon nanotube (HNBR_CB/silica/MWCNT), rubber-rubber hybrid (HNBR and fluororubber (HNBR-FKM)) and rubber-thermoplastic blend (HNBR and cyclic butylene terephthalate oligomers (HNBR-CBT)). The dominant wear mechanisms were investigated by scanning electron microscopy (SEM), and analyzed as a function of composition and testing conditions. Differential scanning calorimetry (DSC), dynamic-mechanical thermal analysis (DMTA), atomic force microscopy (AFM), and transmission electron microscopy (TEM) along with other auxiliary measurements, were adopted to determine the phase structure and network-related properties of the rubber systems. The changes of the friction and wear as a function of type and amount of the additives were explored. The friction process of selected rubbers was also modelled by making use of the finite element method (FEM). The results show that incorporation of filler enhanced generally the wear resistance, hardness, stiffness (storage modulus), and apparent crosslinking of the related rubbers (EPDM-, HNBR- and HNBR-FKM based ones), but did not affect their glass transition temperature. Filling of rubbers usually reduced the coefficient of friction (COF). However, the tribological parameters strongly depended also on the test set-up and test duration. High wear loss was noticed for systems showing the occurrence of Schallamach-type wavy pattern. The blends HNBR-FKM and HNBR-CBT were two-phase structured. In HNBR-FKM, the FKM was dispersed in form of large microscaled domains in the HNBR matrix. This phase structure did not change by incorporation of MWCNT. It was established that the MWCNT was preferentially embedded in the HNBR matrix. Blending HNBR with FKM reduced the stiffness and degree of apparent crosslinking of the blend, which was traced to the dilution of the cure recipe with FKM. The coefficient of friction increased with increasing FKM opposed to the expectation. On the other hand, the specific wear rate (Ws) changed marginally with increasing content of FKM. In HNBR-CBT hybrids the HNBR was the matrix, irrespective to the rather high CBT content. Both the partly and mostly polymerized CBT ((p)CBT and pCBT, respectively) in the hybrids worked as active filler and thus increased the stiffness and hardness. The COF and Ws decreased with increasing CBT content. The FEM results in respect to COF achieved on systems possessing very different structures and thus properties (EPDM_30CB, HNBR-FKM 100-100 and HNBR-(p)CBT 100-100, respectively) were in accordance with the experimental results. This verifies that FEM can be properly used to consider the complex viscoelastic behaviour of rubber materials under dry rolling condition.
Within the last decades, a remarkable development in materials science took place -- nowadays, materials are not only constructed for the use of inert structures but rather designed for certain predefined functions. This innovation was accompanied with the appearance of smart materials with reliable recognition, discrimination and capability of action as well as reaction. Even though ferroelectric materials serve smartly in real applications, they also possess several restrictions at high performance usage. The behavior of these materials is almost linear under the action of low electric fields or low mechanical stresses, but exhibits strong non-linear response under high electric fields or mechanical stresses. High electromechanical loading conditions result in a change of the spontaneous polarization direction with respect to individual domains, which is commonly referred to as domain switching. The aim of the present work is to develop a three-dimensional coupled finite element model, to study the rate-independent and rate-dependent behavior of piezoelectric materials including domain switching based on a micromechanical approach. The proposed model is first elaborated within a two-dimensional finite element setting for piezoelectric materials. Subsequently, the developed two-dimensional model is extended to the three-dimensional case. This work starts with developing a micromechanical model for ferroelectric materials. Ferroelectric materials exhibit ferroelectric domain switching, which refers to the reorientation of domains and occurs under purely electrical loading. For the simulation, a bulk piezoceramic material is considered and each grain is represented by one finite element. In reality, the grains in the bulk ceramics material are randomly oriented. This property is taken into account by applying random orientation as well as uniform distribution for individual elements. Poly-crystalline ferroelectric materials at un-poled virgin state can consequently be characterized by randomly oriented polarization vectors. Energy reduction of individual domains is adopted as a criterion for the initiation of domain switching processes. The macroscopic response of the bulk material is predicted by classical volume-averaging techniques. In general, domain switching does not only depend on external loads but also on neighboring grains, which is commonly denoted as the grain boundary effect. These effects are incorporated into the developed framework via a phenomenologically motivated probabilistic approach by relating the actual energy level to a critical energy level. Subsequently, the order of the chosen polynomial function is optimized so that simulations nicely match measured data. A rate-dependent polarization framework is proposed, which is applied to cyclic electrical loading at various frequencies. The reduction in free energy of a grain is used as a criterion for the onset of the domain switching processes. Nucleation in new grains and propagation of the domain walls during domain switching is modeled by a linear kinetics theory. The simulated results show that for increasing loading frequency the macroscopic coercive field is also increasing and the remanent polarization increases at lower loading amplitudes. The second part of this work is focused on ferroelastic domain switching, which refers to the reorientation of domains under purely mechanical loading. Under sufficiently high mechanical loading, however, the strain directions within single domains reorient with respect to the applied loading direction. The reduction in free energy of a grain is used as a criterion for the domain switching process. The macroscopic response of the bulk material is computed for the hysteresis curve (stress vs strain) whereby uni-axial and quasi-static loading conditions are applied on the bulk material specimen. Grain boundary effects are addressed by incorporating the developed probabilistic approach into this framework and the order of the polynomial function is optimized so that simulations match measured data. Rate dependent domain switching effects are captured for various frequencies and mechanical loading amplitudes by means of the developed volume fraction concept which relates the particular time interval to the switching portion. The final part of this work deals with ferroelectric and ferroelastic domain switching and refers to the reorientation of domains under coupled electromechanical loading. If this free energy for combined electromechanical loading exceeds the critical energy barrier elements are allowed to switch. Firstly, hysteresis and butterfly curves under purely electrical loading are discussed. Secondly, additional mechanical loads in axial and lateral directions are applied to the specimen. The simulated results show that an increasing compressive stress results in enlarged domain switching ranges and that the hysteresis and butterfly curves flatten at higher mechanical loading levels.
A general framework for the thermodynamics of open systems is developed in the spatial and the material setting. Special emphasis is placed on the balance of mass which is enhanced by additional source and flux terms. Different solution strategies within the finite element technique are derived and compared. A number of numerical examples illustrates the features of the proposed approach.