Qualitative and quantitative properties of the dynamics of screw dislocations

This note collects some results on the behaviour of screw dislocation in an elastic medium. By using a semi-discrete model, we are able to investigate two specific aspects of the dynamics, namely (i) the interaction with free boundaries and collision events and (ii) the confinement inside the domain when a suitable Dirichlet-type boundary condition is imposed. In the first case, we analytically prove that free boundaries attract dislocations and we provide an expression for the Peach--Koehler force on a dislocation near the boundary. Moreover, we use this to prove an upper bound on the collision time of a dislocation with the boundary, provided certain geometric conditions are satisfied. An upper bound on the collision time for two dislocations with opposite Burgers vectors hitting each other is also obtained. In the second case, we turn to domains whose boundaries are subject to an external stress. In this situation, we prove that dislocations find it energetically favourable to stay confined inside the material instead of getting closer to the boundary. The result first proved for a single dislocation in the material is extended to a system of many dislocations, for which the analysis requires the careful treatments of the interaction terms.Comment: 12 pages, submitted for the Proceedings volume of the XXIII Conference AIMETA (The Italian Association of Theoretical and Applied Mechanics

Global minimizers for axisymmetric multiphase membranes

We consider a Canham-Helfrich-type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham-Helfrich energy, in which the bending rigidities and spontaneous curvatures are now phase-dependent, and a line tension penalization for the phase interfaces. By restricting attention to axisymmetric surfaces and phase distributions, we extend our previous results for a single phase (arXiv:1202.1979) and prove existence of a global minimizer.Comment: 20 pages, 3 figure

Renormalized Energy and Peach-K\"ohler Forces for Screw Dislocations with Antiplane Shear

We present a variational framework for studying screw dislocations subject to antiplane shear. Using a classical model developed by Cermelli and Gurtin, methods of Calculus of Variations are exploited to prove existence of solutions, and to derive a useful expression of the Peach-K\"ohler forces acting on a system of dislocation. This provides a setting for studying the dynamics of the dislocations, which is done in a forthcoming work.Comment: 22 page

Properties of screw dislocation dynamics: time estimates on boundary and interior collisions

In this paper, the dynamics of a system of a finite number of screw dislocations is studied. Under the assumption of antiplane linear elasticity, the two-dimensional dynamics is determined by the renormalised energy. The interaction of one dislocation with the boundary and of two dislocations of opposite Burgers moduli are analysed in detail and estimates on the collision times are obtained. Some exactly solvable cases and numerical simulations show agreement with the estimates obtained.Comment: 25 pages, 4 figure

Self-propulsion in viscous fluids through shape deformation

In this thesis we address the problem of modeling swimming in viscous fluids. This is a fancy way to denote a fluid dynamics problem in which a deformable object is capable to advance in a low Reynolds number flow governed by the Stokes equations. The fluid is infinitely extended around the swimming body and the propulsive viscous force and torque are those generated by the fluid-swimmer interaction. No-slip boundary conditions are imposed: the velocity of the fluid and that of the swimmer are the same at the contact surface. Moreover, a self-propulsion constraint is enforced: no external forces or torques. The problem is treated with techniques coming from the Calculus of Variations and Continuum Mechanics, through which it is possible to define the coefficients of the ordinary differential equations that govern the position and orientation parameters of the swimmer. In a three-dimensional setting, there are six of them. Conversely, the shape of the swimmer undergoes an infinite-dimensional control. The relations between the infinite-dimensional freely adjustable shape and the six position and orientation variables is given by an explicit linear relation between viscous forces and torques, on one side, and linear and angular velocities on the other. Suitable function spaces are defined to let the variational techniques work, both in the case of a plain viscous fluid (governed by the Stokes system) and in the case of a particulate fluid, which we model using the Brinkman equation. Finally, a control problem for a mono-dimensional swimmer in a viscous fluid is addressed. In this part, which is still work in progress, the existence of an optimal swimming strategy is proved, and the controllability of the swimmer is achieved by showing and explicit sequence of moves to advance. At the very last, the Euler equation for characterizing the optimal chape change is set up, and some comments on its structure are made

Boundary Behavior and Confinement of Screw Dislocations

In this note we discuss two aspects of screw dislocations dynamics: their behavior near the boundary and a way to confine them inside the material. In the former case, we obtain analytical results on the estimates of collision times (one dislocation with the boundary and two dislocations with opposite Burgers vectors with each other); numerical evidence is also provided. In the latter, we obtain analytical results stating that, under imposing a certain type of boundary conditions, it is energetically favorable for dislocations to remain confined inside the domain