Factorization method for second order functional equations

We apply general difference calculus in order to obtain solutions to the functional equations of the second order. We show that factorization method can be successfully applied to the functional case. This method is equivariant under the change of variables. Some examples of applications are presented.Comment: 22 pages, examples and new section added, several correction

Joining ultrafine grained aluminium by friction stir welding – processing, microstructure and mechanical properties

Nowadays different processes are used to improve mechanical properties of materials. In metallic materials, grain size refinement down to nanoscale is one of the most efficient strengthening mechanisms, as predicted by Hall-Petch relationship. Such microstructure refinement can be obtained in several ways, among others by severe plastic deformation (SPD). Although a tremendous progress has been made in the development of SPD methods, the main drawback is the restriction in billet dimensions. The most common shape – rods, are manufacturing with diameter about few or sometimes over a dozen millimetres. Incremental ECAP is a novel tool to manufacture plates with ultrafine grained structure. Possible sizes of plates are promising for future applications, e.g. in automotive industry. Another issue related to ultrafine grained materials is joining without losing their properties governed by the nanoscale structure. Traditional methods cause grain coarsening which is highly unwanted. In this work plates from Al 1050 after Incremental ECAP were joined using Friction Stir Welding. The quality of joints was determined using microscopic observations. Also, the structure of joints and base materials was investigated by light microscopy and transmission electron microscopy. Mechanical properties were measured by microhardness and tensile tests. To investigate mechanical properties like yield strength and tensile strength mini samples were used. Samples were separately cut from the joints and initial materials as well. It allowed to investigate the differences in both areas. The results revealed that joints zone are characterized by lower values of microhardness and tensile properties compared to base materials. Structure investigation showed changes in grain sizes caused by joining process