Zero-range potential in the model of the spinor field interacting with a material plane: conditions for transparency of the plane to Dirac particles

In the model of interaction of a spinor field with a homogeneous isotropic material plane, the possibility of the passage of a Dirac particle through it without reflection and a change in polarization isstudied. The model was proposed earlier in the framework of the Symanzik approach to describe the interaction of spinor quantum fields with two-dimensional materials. In its framework the scattering processes of Dirac particles and the properties of their bound states were investigated. In this paper, we study the conditions under which the plane is completely transparent to the particles. A description of the region of model parameters characterizing the properties of the plane material is obtained explicitlyand given in the article, as well as expressions in terms of these parameters for the momentum and polarization of the particle at which interaction with the plane does not affect its movement.

The Optimal Shape of a Hanging Heavy Body

We consider a heavy body hanging from a fixed point. The weight of the body is known but not its shape. The aim is to find the optimal shape of the body that minimizes the stress concentrations. The body is assumed to be plane and composed of a linearly elastic homogeneous isotropic material. The result is that the optimal boundary, chosen in the class of epitrochoids having the same area, is the epicycloidal cardioid having only one cusp at the hanging point. In technical application, the above results suggest adopting economical shapes in the construction of cylindrical solids, suspended to cables.

Extension and Torsion of Elastic Bars With Initial Twist

In contrast to straight bars, elastic bars with initial twist exhibit coupling between tension and torsion. The coupling is examined for pretwisted bars composed of homogeneous isotropic material under the assumption that each section undergoes the same deformation. The expressions for the axial force and torsional moment are transformed by use of the governing equations and boundary conditions to forms that are particularly useful for rods with small amounts of pretwist. The elliptical section with center on the axis of pretwist and the off-centered circular section are treated in order to illustrate the coupling between extension and torsion caused by the initial twist.