Plasticity as the Γ-limit of a Dislocation Energy

In this thesis, we derive macroscopic crystal plasticity models from mesoscopic dislocation models by means of Γ-convergence as the interatomic distance tends to zero. Crystal plasticity is the effect of a crystal undergoing an irreversible change of shape in response to applied forces. At the atomic scale, dislocations --- which are local defects of the crystalline structure --- are considered to play a main role in this effect. We concentrate on reduced two-dimensional models for straight parallel edge dislocations. Firstly, we consider a model with a nonlinear, rotationally invariant elastic energy density with mixed growth. Under the assumption of well-separateness of dislocations, we identify all scaling regimes of the stored elastic energy with respect to the number of dislocations and prove Γ-convergence in all regimes. As the main mathematical tool to control the non-convexity induced by the rotational invariance of the energy, we prove a generalized rigidity estimate for fields with non-vanishing curl. For a given function with values in the set of 2x2 matrices, the estimate provides a quantitative bound for the distance to a specific rotation in terms of the distance to the set of rotations and the curl of the function. The most important ingredient for the proof is a fine estimate which shows that in two dimensions an integrable vector-valued function f can be decomposed into two parts belonging to certain negative Sobolev spaces with critical exponent such that corresponding estimates depend only on div f and the integral of |f|. This is a generalization of an estimate due to Bourgain and Br'ezis. Secondly, we consider a dislocation model in the setting of linearized elasticity. The main difference to the first case above and existing literature is that we do not assume well-separateness of dislocations. In order to prove meaningful lower bounds, we adapt ball construction techniques which have been used successfully in the context of the Ginzburg-Landau functional. The building block for this technique are good lower bounds on annuli. In contrast to the vortices in the Ginzburg-Landau model, in the setting of linear elasticity, a massive loss of rigidity can be observed on thin annuli which leads to inadequate lower bounds. Hence, our analysis focuses on finding thick annuli which carry almost all relevant energy

Energy Scaling Law for a Singularly Perturbed Four-Gradient Problem in Helimagnetism

We study pattern formation inmagnetic compounds near the helimagnetic/ferromagnetic transition point in case of Dirichlet boundary conditions on the spin field. The energy functional is a continuum approximation of a J1 − J3 model and was recently derived in Cicalese et al. (SIAM J Math Anal 51: 4848–4893, 2019). It contains two parameters, one measuring the incompatibility of the boundary conditions and the other measuring the cost of changes between different chiralities.We prove the scaling law of the minimal energy in terms of these two parameters. The constructions from the upper bound indicate that in some regimes branching-type patterns form close to the boundary of the sample.Peer Reviewe

WWU Soil Ecology Lab Intern

The focus of the research is on how tilling in perennial raspberry fields impacts mycorrhizal fungi in those fields, which in turn increases the health of the raspberry plants in those fields

Visualizing Animal Fire Responses

The following document contains an assembly of 21 pie charts depicting proportions of animal during-fire responses across ecosystems and groups of specific animals within those ecosystems. For further clarification, a good example would be the Coastal Redwoods Ecosystem Pie Charts, which depict a general chart for all animal fire responses before displaying bird fire responses and mammal fire responses in separate charts. The pie charts were originally created for a group ENVS 429 capstone project before being modified based on accessibility guidelines and Edward Tufte’s graphic design principles. This project and its ENVS 429 predecessor are valuable because animal fire responses are relatively under researched, which in turn poses significant land management and policy challenges. While both projects have significant limitations as they stand, further development of either could eventually produce a resource that can help land managers, policymakers, and other working professionals better understand how animals in different ecosystems react to fire. What makes this project particularly significant is that it presents this information in as accessible of a manner as possible, utilizing graphic design principles and accessibility guidelines to make the information of each chart as legible as possible

A System Dynamics Model of the Air Transport System

In this report, we give a complete algebraic description of a system dynamics model of the air transport system, developed to assess the impact of different policies on the adoption rate of fully electric aircraft until the year 2050. Our model consists of the interaction between three major segments, namely air travel demand, airline industry and aircraft manufacturers. This model was used in the paper “How much can electric aircraft contribute to reaching the Flightpath 2050 CO2 emissions goal? A system dynamics approach for European short haul flights” for the computational results therein

A BVBV functional and its relaxation for joint motion estimation and image sequence recovery

The estimation of motion in an image sequence is a fundamental task in image processing. Frequently, the image sequence is corrupted by noise and one simultaneously asks for the underlying motion field and a restored sequence. In smoothly shaded regions of the restored image sequence the brightness constancy assumption along motion paths leads to a pointwise differential condition on the motion field. At object boundaries which are edge discontinuities both for the image intensity and for the motion field this condition is no longer well defined. In this paper a total-variation type functional is discussed for joint image restoration and motion estimation. This functional turns out not to be lower semicontinuous, and in particular fine-scale oscillations may appear around edges. By the general theory of vector valued $BV$ functionals its relaxation leads to the appearance of a singular part of the energy density, which can be determined by the solution of a local minimization problem at edges. Based on bounds for the singular part of the energy and under appropriate assumptions on the local intensity variation one can exclude the existence of microstructures and obtain a model well-suited for simultaneous image restoration and motion estimation. Indeed, the relaxed model incorporates a generalized variational formulation of the brightness constancy assumption. The analytical findings are related to ambiguity problems in motion estimation such as the proper distinction between foreground and background motion at object edges.Comment: 33 pages, 7 figure