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Axisymmetric extrusion and drawing through a die of arbitrary shape is analyzed by the Orowan method. The stress state in a representative element is determined by solving a boundary value problem of plasticity theory.

Discontinuities such as fractures or cracks are common in geo-materials. Connectivity is a comprehensive parameter that includes the influence of length, orientation and density of the discontinuities. The main objectives of this paper are to define connectivity based on the statistical geometrical parameters of a network of random discontinuities and to quantify the hydraulic properties of the network using connectivity. An analytical method is first proposed to evaluate connectivity through the density of degree of freedom (DDOF). The analytically estimated DDOF is verified using numerical analysis. DDOF increases with the increases in the mean length, relative orientation and density of the random discontinuity network. Connectivity becomes better and the permeability becomes larger with increasing DDOF. The representative element volume (REV) of a random discontinuity network can be estimated based on connectivity. When a discontinuity network is poorly connected, a REV does not exist when the size of the discontinuity network is smaller than 10 times of the mean length of the discontinuities. When a discontinuity network is normally connected, a REV exists when the REV size is smaller than 10 times of the mean length of the discontinuities. When the discontinuity network is highly connected, the REV size is only 2–4 times of the mean length of the discontinuities. The results indicate that DDOF is a good indicator to quantify the connectivity of a random discontinuity network and the relationship between connectivity and permeability.

A model and design model for representative-element analysis of the decay of the Saint-Venant edge effects in laminated materials are proposed. The decay of the edge effect in a matrix with laminated covering (a composite material of irregular structure) is considered for the case of symmetric deformation. The source of the edge effect is simulated by a piecewise-constant surface periodic load normal to the layers. This load is local for the working domain and is varied within a boundary section that is commensurable with the characteristic size of the heterogeneity of the material structure. The equations of the linear theory of elasticity, a model of a piecewise-homogeneous medium, and quantitative criteria for identification of edge effects are used. A discrete model of the problem and its solution are constructed within the concept of basic schemes. Data on the zone of the edge effect and the character of its decay for a representative element of the material are analyzed.

This paper is devoted to the discussion of transitive uncertainty mapping in general approximation space. It is proved that the best low-approximation mapping exist if the uncertainty mapping is transitive. Furthermore, the best low-approximation mapping is defined and its properties are discussed.

We show herein that a pattern based on FGLM techniques can be used for computing Grobner bases, or related structures, associated to linear codes. This Grobner bases setting turns out to be strongly related to the combinatorics of the codes.