Patrick W. Dondl

Pinning of interfaces in random media

For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as Quenched Edwards–Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though the local evolution law is viscous (in particular, the velocity of the interface in the model is linear in the driving force).

Confined Elastic Curves

We consider the problem of minimizing Eulers elastica energy for simple closed curves confined to the unit disk. We approximate a simple closed curve by the zero level set of a function with values

Relaxation of the single-slip condition in strain-gradient plasticity

+1

Phase Field Models for Thin Elastic Structures with Topological Constraint

on the inside and

Energy Estimates, Relaxation, and Existence for Strain-Gradient Plasticity with Cross-Hardening

-1

Microstructure in Plasticity, a Comparison between Theory and Experiment

on the outside of the curve. The outer container now becomes just the domain of the phase field. Diffuse approximations of the elastica energy and the curve length are well known; implementing the topological constraint thus becomes the main difficulty here. We propose a solution based on a diffuse approximation of the winding number, present a proof that one can approximate a given sharp interface using a sequence of phase fields, and show some numerical results using finite elements based on subdivision surfaces.

Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients

We consider the variational formulation of both geometrically linear and geometrically nonlinear elasto-plasticity subject to a class of hard single-slip conditions. Such side conditions typically render the associated boundary-value problems non-convex. We show that, for a large class of non-smooth plastic distortions, a given single-slip condition (specification of Burgers vectors) can be relaxed by introducing a microstructure through a two-stage process of mollification and lamination. The relaxed model can be thought of as an aid to simulating macroscopic plastic behaviour without the need to resolve arbitrarily fine spatial scales.

Uniform regularity and convergence of phase-fields for Willmore’s energy.

This article is concerned with the problem of minimising the Willmore energy in the class of connected surfaces with prescribed area which are confined to a small container. We propose a phase field approximation based on De Giorgi’s diffuse Willmore functional to this variational problem. Our main contribution is a penalisation term which ensures connectedness in the sharp interface limit. The penalisation of disconnectedness is based on a geodesic distance chosen to be small between two points that lie on the same connected component of the transition layer of the phase field. We prove that in two dimensions, sequences of phase fields with uniformly bounded diffuse Willmore energy and diffuse area converge uniformly to the zeros of a double-well potential away from the support of a limiting measure. In three dimensions, we show that they converge

The Effect of Forest Dislocations on the Evolution of a Phase-Field Model for Plastic Slip

{\mathcal{H}^1}