Subdifferential Decomposition of 1D-regularized Total Variation with Nonhomogeneous Coefficients

In this paper, we consider a convex function defined as a 1D-regularized total variation with nonhomogeneous coefficients, and prove the Main Theorem concerned with the decomposition of the subdifferential of this convex function to a weighted singular diffusion and a linear regular diffusion. The Main Theorem will be to enhance the previous regularity result for quasilinear equation with singularity, and moreover, it will be to provide some useful information in the advanced mathematical studies of grain boundary motion, based on KWC type energy.

Regularities of Structural Rearrangements in Single- and Bicrystals Near the Contact Zone

The chapter is devoted to the analysis of the features of local structural rearrangementsin nanostructured materialsunder shear loadingand nanoindentation. The study was carried out using molecular dynamics-based computer simulation. In particular, we investigated the features of symmetric tilt grain boundary migration in bcc and fcc metals under shear loading. The main emphasis was on identifying atomic mechanisms responsible for the migration of symmetric tilt grain boundaries. We revealed that grain boundaries of this type can move with abnormally high velocities up to several hundred meters per second. The grain boundary velocity depends on the shear rate and grain boundary structure. It is important to note that the migration of grain boundary does not lead to the formation of structural defects. We showed that grain boundary moves in a pronounced jump-like manner as a result of a certain sequence of self-consistent displacements of grain boundary atomic planes and adjacent planes. The number of atomic planes involved in the migration process depends on the structure of the grain boundary. In the case of bcc vanadium, five planes participate in the migration of the Σ5(210)[001] grain boundary, and three planes determine the Σ5(310)[001] grain boundary motion. The Σ5(310)[001] grain boundary in fcc nickel moves as a result of rearrangements of six atomic planes. The stacking order of atomic planes participating in the grain boundary migration can change. A jump-like manner of grain boundary motion may be divided into two stages. The first stage is a long time interval of stress increase during shear loading. The grain boundary is motionless during this period and accumulates elastic strain energy. This is followed by the stage of jump-like grain boundary motion, which results in rapid stress drop. The related study was focused on understanding the atomic rearrangements responsible for the nucleation of plasticity near different crystallographic surfaces of fcc and bcc metals under nanoindentation. We showed that a wedge-shaped region, which consists of atoms with a changed symmetry of the nearest environment, is formed under the indentation of the (001) surface of the copper crystallite. Stacking faults arise in the (111) atomic planes of the contact zone under the indentation of the (011) surface. Their escape on the side free surface leads to a step formation. Indentation of the (111) surface is accompanied by nucleation of partial dislocations in the contact zone subsequent formation of nanotwins. The results of the nanoindentation of bcc iron bicrystal show that the grain boundary prevents the propagation of structural defects nucleated in the contact zone into the neighboring grain.

Electrochemical drag effect on grain boundary motion in ionic ceramics

The effects of drag imposed by extrinsic ionic species and point defects on the grain boundary motion of ionic polycrystalline ceramics were quantified for the generality of electrical, chemical, or structural driving forces. In the absence of, or for small driving forces, the extended electrochemical grain boundary remains pinned and symmetrically distributed about the structural interface. As the grain boundary begins to move, charged defects accumulate unsymmetrically about the structural grain boundary core. Above the critical driving force for motion, grain boundaries progressively shed individual ionic species, from heavier to lighter, until they display no interfacial electrostatic charge and zero Schottky potential. Ionic p–n junction moving grain boundaries that induce a finite electrostatic potential difference across entire grains are identified for high velocity grains. The developed theory is demonstrated for Fe-doped SrTiO3. The increase in average Fe concentration and grain boundary crystallographic misorientation enhances grain boundary core segregation and results in thick space charge layers, which leads to a stronger drag force that reduces the velocity of the interface. The developed theory sets the stage to assess the effects of externally applied fields such as temperature, electromagnetic fields, and chemical stimuli to control the grain growth for developing textured, oriented microstructures desirable for a wide range of applications.