Vol 15, No 6 (2019)

In the book of R.M. Christensen, “Introduction to the Mechanics of Composites” (1982), a calculation formula is given for the bulk module of polydisperse composites with spherical inclusions. This formula has been known to the Russianspeaking reader for almost 40 years, but unfortunately, it is not used in the practice of building materials science. To identify applied possibilities, R.M. Christensen's formula is modified and reduced to a dimensionless function k = k ( w , η, θ), which depends on three dimensionless parameters, i.e., it depends on three quantities: w is the volume fraction of the inclusion, η - the ratio of the shear modulus of the matrix material to the volume modulus of the same matrix, θ is the ratio of the volume moduli of the matrix materials and inclusion. Numerical studies of this function reveal that in two-phase granular composites, the range of effective moduli is significantly narrowed compared to the region limited by Voigt and Reuss estimates (in the sense of the upper and lower bounds of real values). At the same time, the lower Christensen score is the same as the Reuss score. Numerical and graphically presented results are given on the examples of the study of two characteristic groups of composite materials. In addition, the dimensionless form of the effective module allows to construct a system of visual graphic dependencies of the functions k ( w ) in a flat space k - w . For different values of θ, the function k = k ( w , η) displays a bunch of curved segments, which sets the position of the plane figure in flat space. Examples of constructing figures for characteristic regions of the values of the function k (η, θ, w ) are given.

Relevance. The choice of adequate models of materials and deformation diagrams is of great importance when performing structural calculations in a nonlinear setting. Since there are no instructions on how to use the deformation diagrams of concrete and reinforcement when working together, given in SP 63.13330.2018, it is necessary to introduce assumptions for modeling reinforced concrete structures with finite elements of the same type. The aims of the work are to conduct numerical experiments on testing concrete cylinders for uniaxial compression and to verify the results with normative data. Methods. Numerical experiments were performed in the LS-DYNA software package. This program complex allows to simulate the joint work of concrete and reinforcement with the help of volume (for concrete) and rod (for reinforcement) finite elements. A cylinder with a diameter of 150 mm and a height of 300 mm was taken as model. Samples were modeled by volumetric finite elements. The CSCM - Continuous Surface Cap Model is a nonlinear material used to model concrete. Tests were carried out with samples of the following classes of concrete for cylindrical compressive strength: C12, C16, C20, C25, C30, C35, C40, C45, C50, C55. This corresponds to the following classes of cubic compressive strength: B15, B20, B25, B30, B37, B45, B50, B55, B60, B67. Results. The conducted researches have shown that the character of destruction of samples at numerical experiment corresponds to the character of destruction at tests. The investigated concrete model CSCM can be used in the calculation of concrete and reinforced concrete structures for the main classes of concrete, when taking into account the transition from cubic to prismatic strength and additional correction factors to cylindrical strength.

Relevance. Architects and engineers, designing shells of revolution, use in their projects, as a rule, spherical shells, paraboloids, hyperboloids, and ellipsoids of revolution well proved themselves. But near hundreds of other surfaces of revolution, which can be applied with success in building and in machine-building, are known. Methods. Optimization problem of design of axisymmetric shell subjected to given external load is under consideration. As usual, the solution of this problem consists in the finding of shape of the meridian and in the distribution of the shell thickness along the meridian. In the paper, the narrower problem is considered. That is a selection of the shell shape from several known types, the middle surfaces of which can be given by parametrical equations. The results of static strength analyses of the domes of different Gaussian curvature with the same overall dimensions subjected to the uniformly distributed surface load are presented. Variational-difference energy method of analysis is used. Results. Comparison of results of strength analyses of six selected domes showed that a paraboloid of revolution and a dome with a middle surface in the form of the surface of rotation of the z = - a cosh( x/b ) curve around the Oz axis have the better indices of stress-strain state. These domes work almost in the momentless state and it is very well for thin-walled shell structures. New criterion of optimality can be called “minimum normal stresses in shells of revolution with the same overall dimensions, boundary conditions, and external load”.

Relevance. At present the application of numerical analyses to real problems of dam engineering has suffered at times from the gaps between the specialists of mathematical modeling and dam engineers and managers. The first group usually includes information system specialists because they are able to develop the computer models to their full potential. The professionals belonging to the second group often prefer to revert to traditional methods of calculation and empirical methods based on their proven experience. The aim of the work - based on recommendations of International workshops seminars, organized by the ICOLD Committee on Computational Aspects of Dam Analysis and Design, help dam engineers to interact with mathematical modeling specialists and to work with them without language barriers or gaps in knowledge. In this relation the assessment of reliability and applicability of numerical analyses of dams allows engineers to develop the optimal dam design. Methods. Assessment of the reliability of numerical methods of analyses of dam behavior was based on data of 10 International benchmark-workshop seminars, organized by the Committee in Italy (1991 and 1992), France (1994 and 2009), Spain (1996), USA (1999), Austria (2001), Romania (2003), China (2005), Russia (2007), in which specialists of these countries also took part.

The aim of the work. Free, transverse vibrations are considered heterogeneous along the three spatial coordinates of rectangular plates lying on an inhomogeneous viscoelastic base. It is assumed that the boundary conditions are homogeneous. A closed solution for the problem of free vibration of an inhomogeneous rectangular orthotropic plate based on an inhomogeneous viscoelastic foundation is developed in the article. Young's moduli and the density of the orthotropic plate continuously change with respect to three spatial coordinates, while the characteristics of a viscoelastic base change depending on the coordinates in the plane. Methods. The corresponding equation of motion is obtained using the classical theory of plates. The solution to the problem was constructed using the method of separation of variables and the Bubnov - Galerkin method. Results. Explicit formulas of the fundamental tone of the frequency of the transverse vibration of an anisotropic plate lying on an inhomogeneous viscoelastic base are determined. The influence of heterogeneity of orthotropic materials, viscosity inhomogeneities, inelastic and elastic substrates at dimensionless plate frequencies have been studied in detail.

The aim of the research is to justify in the non-linear statement the overlay principle of fraction creep deformation, known in the linear creep theory as Bolzmann’s principle of superposition. Methods. In contrast to the traditional approach the material of constructive elements is considered as an union of its links with statistical disturbed strength. The model of structural strength allows the deduction of rheological equations. In loading process so called structural stresses of capable to resist links are considered. Results. The modification Bolzmann’s principle of superposition for fraction creep deformations is proposed. This permits its applicability also under non-linearly dependence of deformations on stresses. In according to concept of the statistical distribution of the strengths of links and linear dependence of determinations on structural stresses the rheological of mechanical statement is reduced. This equation implies the suitable on relation problems the linear integral equation. The relation of structural strength of material with its energy of entirety and with the experimentally known independency of specific to strength deformation on age of concrete is showed. The correct interpretations of certain known mechanical state equations for concrete are represented.