Assessment of ease of excavation of surface mines

An empirical ground classification system for surface mines is presented. It is designed to provide a means for determining the ease of excavation, diggability, by mechanical excavators. The background and the methodology for determining this excavation index is introduced. This index, which is suitable for rock characterization relies on the point load index of the rock, its block size as defined by the presence of discontinuities. The system is also flexible to account for the influence of weathering as well as the relative ground structure which may assist or hinder the excavation process. The economic consequences of selecting mining excavation equipment are also addressed. The Excavation Index was applied with success to 49 case studies (Hadjigergiou, J. Studies of machine ground interaction in surface mines. Ph.D. thesis, McGill University, 1993, p. 214).     

Bifurcation of solutions of an impulsive boundary-value problem

We establish constructive conditions for the bifurcation of solutions and construct an iteration procedure for finding solutions of a linear Noether boundary-value problem for a system of ordinary differential equations with pulse action in the critical case. We obtain an estimate for the range of values of a small parameter for which the iteration procedure converges. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 21–31, January–March, 2007.     

Discretization of an admixture flux within the framework of the fractal model of anomalous diffusion

Within the framework of the fractal mobile-immobile medium model describing non-Fickian effects occurring in admixture seepage due to particle adhesion to the solid matrix, an expression for the admixture flux is derived. Flow discretization intended for finite-difference calculations is proposed and used as a basis for a conservation-law scheme for solving the model equations with account for admixture sources. Several one-dimensional test problems of admixture propagation in an imposed seepage flow are solved using the approach developed.     

The re-examination of the weakly nonlinear theory of hydrodynamic stability

The weakly nonlinear theory has been widely applied in the problem of hydrodynamicstability and also in other fields.However,although its application has been successful forsome problems,yet,for other problems,the results obtained are not satisfactory,especiallyfor problems like transition or the evolution of the vortex in the free shear flow,for whichthe goal of the theoretical investigation is not seeking for a steady state,but predicting anevolutional process.In this paper,we shall examine the reason for the unsuccessfulness andsuggest ways for its amendment.     

Investigation of the structure of the set of continuous solutions of systems of linear functional difference equations

We establish conditions for the existence of continuous solutions of systems of linear functional difference equations with linearly transformed argument and develop a method for the construction of these solutions.     

Motion of bed-loads under the influence of flow of a liquid

Equations are obtained for the motion of a water-soil mixture in the layer next to the base. The water-soil mixture is modeled by means of a viscous-friable medium, and the acceleration of the mixture is assumed to be small and is not taken into account. The validity of the equations is confirmed by the experimental data for the following characteristics of uniform motion of a flow: for the speed of the start of particles touching on an even bed and with allowance for inclines of a bed, and also for the flow rate of loads on an even bed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 182–185.The author is grateful to A. G. Kulikovskii, V. Ya. Shkadov, and M. É. Églit for discussing the results of the study.     

Homogenization of the equations of the Cosserat theory of elasticity of inhomogeneous bodies

The paper deals with the homogenization of a boundary value problem for an inhomogeneous body with Cosserat properties, which is referred to as the original problem. The homogenization process is understood as a method for representing the solution of the original problem in terms of the solution of precisely the same problem for a body with homogeneous properties. The problem for a body with homogeneous properties is called the accompanying problem, and the body itself, the accompanying homogeneous body. As a rule, a constructive homogenization procedure includes the following three stages: at the first stage, the properties of the inhomogeneous body are used to find the properties of the accompanying homogeneous body (efficient properties); at the second stage, the boundary value problem is solved for the accompanying body; at the third stage, the solution of the accompanying problem is used to find the solution of the original problem. This approach was implemented in mechanics of composite materials constructed of numerous representative elements. A significant contribution to the development of mechanics of composites is due to Rabotnov [1–3] and his students. Recently, the homogenization method has been widely used to solve problems for composites of regular structure by expanding the solution of the original problem in a power series in a small geometric parameter equal to the ratio of the characteristic dimension of the periodicity cell to the characteristic dimension of the entire body. The papers by Bakhvalov [4–6] and Pobedrya [7] were the first in the field. At present, there are numerous monographs partially or completely dealing with the method of a small geometric parameter [8–14]. Isolated problems for inhomogeneous bodies with nonperiodic dependence of their properties on the coordinates were considered by many authors. Most of such papers published before 1973 are collected in two vast bibliographic indices [15, 16]. General methods were considered, and many specific problems of the theory of elasticity of continuously inhomogeneous bodies were solved in Lomakin’s papers and his monograph [17]. The theory of torsion of inhomogeneous anisotropic rods was considered in [18]. In 1991, in his Doctoral dissertation, one of the authors of this paper proposed a version of the homogenization method based on an integral formula representing the solution of the original static problem of inhomogeneous elasticity via the solution of the accompanying problem [19, 20]. An integral formula for the dynamic problem of elasticity was published somewhat later [21]. This integral formula was used to develop a constructive method for the homogenization of the dynamic problem of inhomogeneous elasticity, which can be used in the case of both periodic and nonperiodic inhomogeneity of the properties [22]. The integral formula in the case of the Cosserat theory of elasticity was published in [23]. The present paper briefly presents constructive methods for homogenizing the problems of the Cosserat theory of elasticity based on the integral formula.     

Estimate of the residual volume of viscoplastic oil after flooding

The problem of calculating regions of residual viscoplastic oil with a critical pressure gradient is considered. This problem has hitherto been solved mainly for uniform strata (see, for example, [1–3]) but in the present paper, which follows the work of Entov, Pankov, and Pan'ko [4], the problem is considered not only for uniform strata but also for layered strata and vertically nonuniform strata. It is shown that for sufficiently thin strata allowing free flow between individual seams the formulation of the problem is the same for the piston model and the Buckley-Leverett model and reduces to a two-dimensional boundary value problem with a continuous nonlinear effective flow law for the water. An approximate numerical method of calculating regions of residual oil that is suitable for the three listed types of strata is described and illustrated by examples.     

Disintegration of axisymmetric body of revolution made of a material of complex chemical composition in a high-enthalpy airflow

We present a numerical solution of the problem of the disintegration in a high enthalpy airflow of an axisymmetric body of revolution made of a plastic of complex chemical composition. The system of partial differential equations for the laminar multicomponent boundary layer is solved by the pivotal method. The use of an experimental kinetic condition for the disintegration in place of the vapor pressure curve, which is usually used for single-component subliming materials, permits the use of a single technique for calculating both coking and noncoking materials of complex composition. Calculated is the composition of the products of combustion of materials consisting of the chemical elements, H, C, N, O. As an example the results are presented of the calculation of the disintegration of a sphere made from a material similar to textolite in composition.The author wishes to thank G. A. Tirskii for guidance in this study and E. A. Gershbeyn for discussions of the solution scheme.     

Study of the anisotropy of sliding stops of mobile robots

The interaction of mini-robot contact elements with the surface is studied. The values of basic parameters necessary for successful operation of such elements are estimated. A procedure for calculating the value of anisotropic friction typical to adhesive material with haired structure is proposed. The results of this paper can be used to design robots intended for motion in bounded spaces, in particular, in narrow tubes.     

Determination of natural frequencies of vibration of a sandwich plate

In this paper, an expression is derived for the natural frequencies of vibration of a simply supported sandwich plate. Experimental procedures and results for the subject problem are presented. The experimental data obtained are in good agreement with the theoretical results. A literature survey^{1, 2} shows that no work has been done on the experimental determination of natural frequencies of vibration of sandwich plates. A general analysis of flexural vibrations of elastic sandwich plates was presented by Yu in Ref. 3. The vibration analysis based upon this theory is, in general, very complicated due to the high order of the equations. It was then simplified⁴ for low-frequency ranges and for ordinary sandwich plates.     

Method of representation of solution of the Blasius equation and its applications

In the articles published on the Blasius problem [1–3] the solutions are investigated either by some numerical method or by joining the series solution for small values of the argument with the asymptotic expansion for large values. In the present article we give a new method of representing the solutions for different boundary-value problems associated with the Blasius equation. The properties of the obtained solution are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–146, May–June, 1975.The author expresses deep gratitude to Yu. A. Dem'yanov for discussion of the work and for valuable comments.     

Method of reverberation ray matrix for static analysis of planar framed structures composed of anisotropic Timoshenko beam members

Based on the method of reverberation ray matrix(MRRM), a reverberation matrix for planar framed structures composed of anisotropic Timoshenko(T) beam members containing completely hinged joints is developed for static analysis of such structures.In the MRRM for dynamic analysis, amplitudes of arriving and departing waves for joints are chosen as unknown quantities. However, for the present case of static analysis, displacements and rotational angles at the ends of each beam member are directly considered as unknown quantities. The expressions for stiffness matrices for anisotropic beam members are developed. A corresponding reverberation matrix is derived analytically for exact and unified determination on the displacements and internal forces at both ends of each member and arbitrary cross sectional locations in the structure. Numerical examples are given and compared with the finite element method(FEM) results to validate the present model. The characteristic parameter analysis is performed to demonstrate accuracy of the present model with the T beam theory in contrast with errors in the usual model based on the Euler-Bernoulli(EB) beam theory. The resulting reverberation matrix can be used for exact calculation of anisotropic framed structures as well as for parameter analysis of geometrical and material properties of the framed structures.     

Invariants of a Remarkable Family of Nonlinear Equations

In classical literature, invariants of families of differentialequations were considered for linear equations only, e.g. the renownedLaplace invariants for linear hyperbolic partial differential equationsand invariants of linear ordinary differential equations with variablecoefficients. The restriction to linear equations was essential inpioneering works of Cockle, Laguerre, Halphen, andForsyth for tackling the problem of invariants of differentialequations. Lie regretted that these authors did not use advantagesprovided by his theory of infinite continuous groups, but he himself didnot undertake further developments in this direction.Recently, the present author considered the possibility hinted byLie's remark and introduced the infinitesimal technique in thetheory of invariants of families of differential equations thatwas lacking in old methods. In consequence, a simple unifiedapproach was developed for calculation of invariants of algebraicand differential equations independent on the assumption oflinearity of equations. It was employed recently for calculationof Laplace type invariants for parabolic equations. Here, themethod is applied to calculation of invariants for the family ofnonlinear equations appearing in the problem on linearization ofnonlinear ordinary differential equations.     

Effect of sequence of combined loading on buckling of stiffened shells

The vibration-correlation technique, VCT for definition of real boundary conditions, and the method of repeated buckling were employed for nondestructive generation of improved interaction curves for buckling of stringer-stiffened circular-cylindrical shells subjected to a combined axial compression and external-pressure state of loading. Thirteen shells were tested, five on clamped boundary conditions and eight on norminal simple supports. The study also included an assessment of the influence of the order of loading on the behavior of the shells before and at buckling as a result of the nonlinear interaction. It has been shown that the VCT and repeated buckling approach are feasible for closely stiffened shells and are adequate tools for the derivation of more realistic buckling interaction curves. It appears that the sequence of loading, constant axial compression first and then increasing the external pressure until buckling occurs, or the reverse order of loading, does not influence the buckling loads.     

Solution of a Problem of Free Vibrations of a Nonclosed Shell of Revolution under Conditions of its Singular Perturbation

We develop a variational method for solving the spectral problem of free vibrations of a shell of revolution nonclosed in the meridional direction. This method is equally efficient for both medium and small values of the relative thickness of the shell. Coordinate systems of functions are constructed with regard for the structure of formal asymptotic expansions of a fundamental system of solutions of the initial equations. As an example, we calculate frequencies and forms of vibrations of a circular cylindrical shell and show that the algorithm proposed for solving the considered problem guarantees the uniform convergence of solutions and their first three derivatives in the entire region of integration of the equations. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 415–432, July–September, 2005.     

Stability of nonlinear local axisymmetrical strains of shells of revolution

A brief review of the results of investigation of the stability of the axisymmetrical strains of elastic shells of revolution is contained in [1, 2]. In [3] the problem was formulated and solved for a round shell, uniformly loaded along its hinged edge by a radial compressive force. Below, this problem is formulated for an arbitrary shell of revolution with a uniformly compressed hinged edge. Results of its solution are given for conical and spherical shells.     

Invariant manifolds of one class of systems of impulsive differential equations

We consider general problems related to the existence of invariant toroidal sets for linear and weakly nonlinear systems of impulsive differential equations defined in the direct product of an m-dimensional torus and an n-dimensional Euclidean space. We investigate classes of problems for which the conditions for the existence of invariant toroidal manifolds are satisfied.     

Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity

Well-posedness results for the state-based peridynamic nonlocal continuum model of solid mechanics are established with the help of a nonlocal vector calculus. The peridynamic strain energy density for an elastic constitutively linear anisotropic heterogeneous solid is expressed in terms of the field operators of that calculus, after which a variational principle for the equilibrium state is defined. The peridynamic Navier equilibrium equation is then derived as the first-order necessary conditions and are shown to reduce, for the case of homogeneous materials, to the classical Navier equation as the extent of nonlocal interactions vanishes. Then, for certain peridynamic constitutive relations, the peridynamic energy space is shown to be equivalent to the space of square-integrable functions; this result leads to well-posedness results for volume-constrained problems of both the Dirichlet and Neumann types. Using standard results, well-posedness is also established for the time-dependent peridynamic equation of motion.