On the Disordered, Collective and Patterned Movements of Media. Qualitative Analysis of Movements

The motion of elements is classified from a topological aspect. We separate three different, disordered, collective and patterned motion forms of media. We will reveal what concrete shapes the different type media take. In case of a limited number of particles, a discrete description is suitable for tracking the mechanic behavior of the medium. In case of the large of a number of particles we apply in some sense continuous mathematical model according to the motion occurring in the medium instead of a discrete description. We postulate a statistical distribution to the disordered motion; we "distribute" the particles for the collective motion in the physical space and apply a differential geometric description. We assign continuous flow lines to the regular patterned motion of the medium with free particles while we assign energy dissipation to the flow image an irregular patterned motion. In the case of deformable solid body built by periodically arranged rigid bodies the state functions with discrete domain of definition are represented by functions with continuous domain of definition. Granular conglomerations seem to be the only such medium, which allow tracking of the state of all granules.Disordered motion leads to thermodynamic - statistical description. The collective motion of free particles leads to the description of the laminar flow of particles, the collective motion of fixed particles leads to the classical continuum. The patterned motion of free particles leads to the description of vortex or turbulent flow, the patterned motion of fixed particles leads to grid continua.

Displacement: translation and rotation. Differences and Similarities in the Discrete and Continuous Models

The motion (displacement) of the Euclidean space can be decomposed into translation and rotation. The two kinds of motion of the Euclidean space based on two structures of the Euclidean space: The first one is the topological structure, the second one is the idea of distance. The motion is such a (topological) map, that the distance of any two points remains the same. The bounded and closed domain of the Euclidean space is taken as a model of the rigid body. The bounded and closed domain of the Euclidean space is also taken as a model of the deformable solid body. The map – i.e. the displacement field – of the deformable solid body is continuous, but is not (necessarily) motion; the size and the shape of body can change. The material has atomic-molecular structure. In compliance with it, the material can be comprehended as a discrete system. In this case the elements of the material, as an atom, molecule, grain, can be comprehended as either material point, or rigid body. In the first case the kinematical freedom is the translation, in the latter case the translation and the rotation. In the paper we analyse how the kinematical behaviour of the discrete and continuous mechanical system can be characterise by translation and rotation. In the discrete system the two motions are independent variable. At the same time they characterise the movement of the body different way. For instance homogeneous local translation gives the global translation, but the homogeneous local rotation does not give the global rotation. To realise global rotation in a discrete system on one hand global rotation of the position of the discrete elements, on the other hand homogeneous local rotations of the discrete elements in harmony with global rotation are required. In the continuous system the two kinds of movement cannot be interpreted: a point cannot rotate, a rotation of surrounding of a point or direction can be interpreted. The kinematical characteristics, as the displacement (practically this is equal to translation) of (neighbourhood of) point, the rotation of surrounding of that point and the rotation of a direction went through that point are not independent variables: the translation of a point determines the rotation of the surrounding of that point as well as the rotation of a direction went through that point. With accordance this statement the displacement (practically translation) (field) as the only kinematical variable can be interpreted in the continuous medium.

Experimental and theoretical study of elastoplastic deformation, loss of stability and postbuckling behavior of cylindrical shells with discrete aggregate during bending

The monograph presents the developed and experimentally substantiated computational models, presents a numerical analysis of nonlinear deformation, loss of stability and postbuckling behavior of the shell structures of large-sized tanks for transporting bulk materials during bending. The book is intended for researchers, teachers, graduate students and students of higher educational institutions specializing in the field of mechanics of a deformable solid body.

Conditions of equivalence of effects for the solid body from incompressible material

Polemicizing with the existing opinion that modern numerical methods allow to solve practically any problem of mechanics, it should be noted that analytical and experimental methods still are relevant, and a complex of methods leads to development of mechanics of a deformable solid body. At present one of the most important directions of development of mechanics of a deformable solid body is creation of the approaches that allow to combine organically great computing opportunities of modern supercomputers with experimental methods of the material and design research. In engineering practice at production of designs and products incompressible materials are widely used. Assessment of their durability requires detailed studying of deflected mode caused by action of various loadings and forces. For a solid body from incompressible material, using the resolving equations set of mechanics of a deformable solid body, at action of the compelled deformations of a general view, volume and superficial forces conditions of equivalence are established. It is shown that the known solutions are special cases of the established equivalence conditions. The efficiency of the analytical solution of a three-dimensional task on the rotating disk from incompressible material is shown by the method of equivalence of effects.