I. Kunin

Kolmogorov complexity and chaotic phenomena

Abstract Born about three decades ago, Kolmogorov complexity theory (KC) led to important discoveries that, in particular, give a new understanding of the fundamental problem: interrelations between classical continuum mathematics and reality (physics, biology, engineering sciences,…). Specifically, in addition to the equations, physicists use the following additional difficult-to-formalize property: that the initial conditions and the value of the parameters must not be abnormal. We will describe a natural formalization of this property, and show that this formalization in good accordance with theoretical physics. At present, this formalization has been mainly applied to the foundations of physics. However, potentially, more practical engineering applications are possible.

On voids of minimum stress concentration

Abstract An isotropic elastic medium containing a void is loaded at infinity by given stresses. The problem of finding a minimizing void shape for the stress concentration is formulated. It is proved that a sufficient condition for a surface to be a minimizer is that the two surface principal stresses be constant and equal. A class of ellipsoids having this property is exhibited and relations between the applied stresses and the ellipsoid parameters are established.

On foundations of the theory of elastic media with microstructure

Abstract A critical analysis of foundations and domain of applicability of the theory of elastic media with microstructure and nonlocal elasticity is given.

An algebra of tensor operators and its applications to elasticity

Abstract An algebra of tensors of order four which depend on a unit vector and the Kronecker delta is investigated. It is shown that the algebra has natural applications in classical elasticity as well as in the continuum theory of dislocations. Explicit formulas for the corresponding Greens functions and projection operators are obtained. It is shown that a remarkable correspondence exists between global projection operators and local ones associated with strain and stress jumps at surfaces of discontinuity. This correspondence affords a unified treatment for global and local projection operators.

On elastic crack-inclusion interaction

Abstract The problem of a crack near an inclusion is treated using the projection integral equation method. An approximation for elastic fields and the energy of interaction is obtained. Relations between J- and M-integrals and those integrals with the interaction energy are established.

Elastic Medium with Random Fields of Inhomogeneities

In this chapter the method of the effective field is applied to solve problems for composites and cracked solids. Under the assumption of a random change of the effective field from one particle to another the formulae for the first and second moments of random stress-strain fields are presented.

Kinematics of media with continuously changing topology

The fundamental postulate of continuum mechanics states that a body is a three-dimensional differentiable manifold and its motions are diffeomorphisms. Simple thought experiments with cyclic motions of dislocations show that they do not preserve topology (set of neighborhoods). The same is valid for chaotic and turbulent motions with coarse-graining. To describe such motions, kinematics of a generalized continuum mechanics is suggested. Observables are defined operationally in the laboratory system which is not anymore equivalent to the Lagrangian picture. The body is a submanifold of a higher-dimensional space and generalized motions are its diffeomorphisms. In a gauge-theoretic interpretation, the motion is a translational connection with the curvature identified as a “dislocation” density-flux.

Inhomogeneous elastic medium with nonlocal interaction

In [1] the author examined a macroscopically homogeneous elastic medium of simple structure with spatial dispersion. In that case the assumption of the existence of an elementary unit of length and long-range forces conditioned the nonlocalizability of the theory, and the macroscopic homogeneity was manifested in the invariance of the integral operators under shear (difference kernels).In this paper the more general model of an inhomogeneous elastic medium of simple structure with nonlocal interaction is constructed. In § 1 the existence of a symmetric stress tensor is proved with broad assumptions, and the corresponding operator Hookes law is written down. As a corollary, the usual expression for the energy density is obtained. In §2 the case of point defects is considered. An explicit expression is found for the Greens tensor for a medium with point defects in terms of the Greens tensor for the homogeneous medium. With the help of the Greens tensor the self-energy of the defect and the energy of the interaction force are calculated.

On extracting physical information from mathematical models of chaotic and complex systems

Abstract The paper describes a multi-structural approach to the problem indicated in the title. In analogy with quantum mechanics, at the core of the approach are two notions: states and (generalized) observables. This motivates us to distinguish between mathematical and physical dynamical systems. The first class deals with mathematical models and states (solutions) only. The second one adds physical realizations and observables, i.e. all methods of extracting useful information. Observables include: renormalization, optimal gauging, covering–coloring, discretization, tensorial measures of chaos, etc. The corresponding algorithms are correlated to Kolmogorov complexity and are intrinsic parts of observables, and thus of physical systems. The approach is illustrated by examples.

Centroidal trajectories and frames for chaotic dynamical systems

Using Lorenz system as an example, we demonstrate how introduction of lattice in phase space can be used to extract new information from chaotic dynamical system.