Scale-Bridging of Elasto-Plastic Microstructures using Statistically Similar Representative Volume Elements

Die vorliegende Arbeit behandelt die numerische Modellierung des mechanischen Verhaltens mikroheterogener Materialien, wobei das Hauptaugenmerk auf Dualphasenstähle gelegt wird. Ihr makroskopisches Verhalten wird durch die Interaktion der Einzelphasen auf mikrostruktureller Ebene geprägt. Der Einfluss der Morphologie einer realistischen Mikrostruktur kann durch die Verwendung von repräsentativen Volumenelementen (RVEs) unter Anwendung der FE²-Methode direkt in die Materialmodellierung einbezogen werden. Dabei entsteht für RVEs, die als Ausschnitte einer realen Mikrostruktur konstruiert werden, ein enormer Rechenaufwand. Eine Reduzierung des Aufwands ist durch die Verwendung von statistisch ähnlichen RVEs (SSRVEs) möglich. Diese sind durch Ähnlichkeit in Bezug auf bestimmte statistische Maße definiert und liefern gleichzeitig Gleichartigkeit des mechanischen Verhaltens. Die verschiedenen Aspekte der Konstruktion von SSRVEs sind ein Schwerpunkt dieser Arbeit. Es wird gezeigt, dass SSRVEs die mechanischen Eigenschaften der realen Mikrostruktur widerspiegeln und damit ihre Verwendung im Rahmen der FE² -Methode ermöglicht wird. Die Simulation makroskopischer Eigenschaften basierend auf polykristallinen RVEs wird gezeigt. Diese ermöglichen die Beschreibung polykristalliner Materialien, welche von ihrer mikrostrukturellen Textur geprägt werden.The present work deals with the numerical modeling of the mechanical behavior of microheterogeneous materials, with a focus on dual-phase steel. The macroscopic behavior of this material is largely influenced by an interaction of the microstructural constituents. The influence of the morphology of a real microstructure can be included in the material modeling by the application of a suitable representative volume element (RVE) in a direct micro-macro homogenization scheme (also known as FE²-method). However, the use of sections of a real microstructure as an RVE can lead to huge computational costs. A cost reduction can be achieved by the application of statistically similar RVEs (SSRVEs). They are governed by similarities of selected statistical measures with respect to a real microstructure and show a comparable mechanical behavior. The different aspects in the construction method are a main focus of this work. It is shown that SSRVEs can resemble the mechanical behavior of a real DP steel microstructure appropriately, which permits their use in FE²-simulations instead of real microstructures. Aiming for a description of polycrystalline materials governed by texture, the simulation of macroscopic properties based on polycrystalline RVEs is shown

Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model

In this paper we model the size-effects of metamaterial beams under bending with the aid of the relaxed micromorphic continuum. We analyze first the size-dependent bending stiffness of heterogeneous fully discretized metamaterial beams subjected to pure bending loads. Two equivalent loading schemes are introduced which lead to a constant moment along the beam length with no shear force. The relaxed micromorphic model is employed then to retrieve the size-effects. We present a procedure for the determination of the material parameters of the relaxed micromorphic model based on the fact that the model operates between two well-defined scales. These scales are given by linear elasticity with micro and macro elasticity tensors which bound the relaxed micromorphic continuum from above and below, respectively. The micro elasticity tensor is specified as the maximum possible stiffness that is exhibited by the assumed metamaterial while the macro elasticity tensor is given by standard periodic first-order homogenization. For the identification of the micro elasticity tensor, two different approaches are shown which rely on affine and non-affine Dirichlet boundary conditions of candidate unit cell variants with the possible stiffest response. The consistent coupling condition is shown to allow the model to act on the whole intended range between macro and micro elasticity tensors for both loading cases. We fit the relaxed micromorphic model against the fully resolved metamaterial solution by controlling the curvature magnitude after linking it with the specimen's size. The obtained parameters of the relaxed micromorphic model are tested for two additional loading scenarios

Construction of 3d statistically similar rves for dual-phase steel microstructures

A method for the construction of 3D Statistically Similar RVEs for dual-phase steel (DP steel) microstructures in presented in this paper. DP steels have enhanced material properties compared to conventional steels which make them favorable for many engineering applications. Since these properties originate from the microstructure of the two-phase material, microstructural effects should be taken into account. This can be achieved by using the FE2 method, however, this method requires RVEs of low complexity in order to end up in calculations with reasonable computing time. Instead of using RVEs as direct substructures of a real microstructure, SSRVEs with less complex inclusion morphology can be constructed, which still represent the mechanical response of the material accurately enough while providing a speedup due to the lower complexity of discretization. The method for the construction of such SSRVEs described here is based on the minimization of a least-square functional taking into account distinct statistical measures computed for the real microstructure and the SSRVE. Here, the focus is on the construction of three-dimensional SSRVEs. The performance of those SSRVEs is shown and an inhomogeneous numerical example using the FE2 method combined with SSRVEs is presented

Lagrange and H(curl⁡,B)H(\operatorname{curl},{\cal B}) based Finite Element formulations for the relaxed micromorphic model

Modeling the unusual mechanical properties of metamaterials is a challenging topic for the mechanics community and enriched continuum theories are promising computational tools for such materials. The so-called relaxed micromorphic model has shown many advantages in this field. In this contribution, we present the significant aspects related to the relaxed micromorphic model realization with the finite element method. The variational problem is derived and different FEM-formulations for the two-dimensional case are presented. These are a nodal standard formulation $H^1({\cal B}) \times H^1({\cal B})$ and a nodal-edge formulation $H^1({\cal B}) \times H(\operatorname{curl}, {\cal B})$, where the latter employs the N\'ed\'elec space. However, the implementation of higher-order N\'ed\'elec elements is not trivial and requires some technicalities which are demonstrated. We discuss the convergence behavior of Lagrange-type and tangential-conforming finite element discretizations. Moreover, we analyze the characteristic length effect on the different components of the model and reveal how the size-effect property is captured via this characteristic length

A computational approach to identify the material parameters of the relaxed micromorphic model

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. The relaxed micromorphic model is a generalized continuum that utilizes the $Curl$ of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature..

A computational approach to identify the material parameters of the relaxed micromorphic model

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. We avoid establishing exact micro–macro transition relations, as in classical homogenization theory, because defining a representative volume element is not feasible in the absence of scale separation, as such an element does not exist. The relaxed micromorphic model is a generalized continuum that utilizes the Curl of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature; one simplified formulation involving only one single characteristic length and a simplified isotropic curvature with three parameters. The relaxed micromorphic model demonstrates good agreement with the fully-resolved heterogeneous solution after optimizing only four parameters. The “simplified” full micromorphic model, which includes isotropic curvature and involves the optimization of seven parameters, does not achieve superior results, while the Cosserat model exhibits the poorest fitting

A computational approach to identify the material parameters of the relaxed micromorphic model

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. The relaxed micromorphic model is a generalized continuum that utilizes the \Curl of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature..

Experimental and numerical investigations of the development of residual stresses in thermo-mechanically processed Cr-alloyed steel 1.3505

Residual stresses in components are a central issue in almost every manufacturing process, as they influence the performance of the final part. Regarding hot forming processes, there is a great potential for defining a targeted residual stress state, as many adjustment parameters, such as deformation state or temperature profile, are available that influence residual stresses. To ensure appropriate numerical modeling of residual stresses in hot forming processes, comprehensive material characterization and suitable multiscale Finite Element (FE) simulations are required. In this paper, experimental and numerical investigations of thermo-mechanically processed steel alloy 1.3505 (DIN 100Cr6) are presented that serve as a basis for further optimization of numerically modeled residual stresses. For this purpose, cylindrical upsetting tests at high temperature with subsequently cooling of the parts in the media air or water are carried out. Additionally, the process is simulated on the macroscale and compared to the results based on the experimental investigations. Therefore, the experimentally processed specimens are examined regarding the resulting microstructure, distortions, and residual stresses. For the investigation on a smaller scale, a numerical model is set up based on the state-data of the macroscopic simulation and experiments, simulating the transformation of the microstructure using phase-field theory and FE analysis on micro- and meso-scopic level

Vicious and virtuous relationships between procrastination and emotions: an investigation of the reciprocal relationship between academic procrastination and learning-related anxiety and hope

Although cross-sectional studies depict (negative) emotions as both antecedents and consequences of trait procrastination, longitudinal studies examining reciprocal relationships between procrastination and emotions are scant. Yet, investigating reciprocal relationships between procrastination and emotions within long-term frameworks can shed light on the mechanisms underlying these relationships. Additionally, the role of positive emotions concerning procrastination is largely unattended to in the procrastination–emotion research; albeit, this perspective can inform preventive and intervention measures against procrastination. In the present study, we explored reciprocal associations between trait academic procrastination on the one hand and trait-like learning-related anxiety and hope on the other hand over one semester. Overall, N = 789 students in German universities participated in a three-wave online panel study. Participants responded to questions on academic procrastination as well as learning-related anxiety and hope at the beginning (T1), middle (T2), and end (T3) of the lecture period of the semester in approximately 6-week measurement intervals. A latent cross-lagged panel model was used to test the hypotheses. After accounting for autoregressive effects, our results showed that academic procrastination at T1 positively predicted learning-related anxiety at T2. In contrast, academic procrastination at T1 negatively predicted learning-related hope at T2, which in turn negatively predicted academic procrastination at T3. Our results highlight positive emotions (e.g., hope) as also significant factors for procrastination and suggest them as possible “protective factors” against procrastination. Boosting positive emotions as part of interventions against procrastination could potentially help reduce the tendency to procrastinate