Models for Material Defects and Grain Boundaries

What it is about

You are dealing with variational “semi-discrete” models for defects in metals and their significance for plastic deformation. The course covers multiscale analyses that lead to the formulation of line-energy models for dislocations and the modeling of grain boundaries. In the end you can apply mathematical techniques to analyze such models and understand the underlying energy estimates and asymptotics.

What you will be able to do

Apply variational and non-variational techniques
Investigate existence and qualitative properties of nonlinear PDE
Understanding of linear dislocation theories in 2D and 3D
Familiarity with geometrically nonlinear models and asymptotics
Insight into rigidity results for incompatible fields and small-angle grain boundaries

What the module consists of

VorlesungDelivery of theoretical content, examples and discussions with students

Teaching method

Vorlesung mit DiskussionContent is presented, illustrated with examples and discussed with students to stimulate independent analysis and literature work
Betreute Übungsaufgaben während der VorlesungStudents work on concrete examples and simple tasks under guidance for deepening and self-check

What you should already know

MA1001/MA0001 Analysis 1recommended
MA1002/MA0002 Analysis 2recommended
MA2003 Measure and Integration Theoryrecommended
MA3001 Functional Analysisrecommended
MA3005 Partial Differential Equationsrecommended

Assessment

Oral exam

Oral examination (45–60 minutes) on selected results from the course or on previously agreed related topics. It is expected that students understand fundamental techniques and can apply them to models of materials science. No aids are specified.

Reading list

L. Ambrosio, N. Fusco, D. Pallara - Functions of Bounded Variation and Free Discontinuity Problems
S. Conti, A. Garroni, M. Ortiz - The line-tension approximation as the dilute limit of linear-elastic dislocations (Arch. Ration. Mech. Anal., 2015)
A. Garroni, G. Leoni, M. Ponsiglione - Gradient theory for plasticity via homogenization of discrete dislocations (J. Eur. Math. Soc., 2010)
G. Lauteri, S. Luckhaus - An energy estimate for dislocation configurations and the emergence of Cosserat-type structures in metal plasticity (Preprint arXiv:1608.06155, 2016)
S. Müller, L. Scardia, C. I. Zeppieri - Geometric rigidity for incompatible fields and an application to strain-gradient plasticity (Indiana Univ. Math. J., 2014)
S. Müller, L. Scardia, C. I. Zeppieri - Gradient theory for geometrically nonlinear plasticity via the homogenization of dislocations (in Analysis and Computation of Microstructure in Finite Plasticity, Springer, 2015)

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Models for Material Defects and Grain Boundaries

What it is about

What you will be able to do

What the module consists of

Teaching method

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