Antiplane deformation of an isotropic massif with a curvilinear cavity

We solve the problem of antiplane deformation for an infinite isotropic massif with a curvilinear cavity in which a harmonic shear wave is propagating. The solution of the problem, which is based on the application of the theory of functions of a complex variable, is reduced to finding unknown constants from a system of linear algebraic equations. Numerical results are given for a circular cavity. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 115–119.     

An analytic method of studying the nonlinearly elastic equilibrium of a rectangular plate of variable thickness under bending

We propose an approximate analytic method of solving three-dimensional boundary-value problems of the physically nonlinear theory of elasticity for thick rectangular plates of variable thickness subject to a transverse load. The method is used to seek a solution of this problem in the form of double power series in small dimensionless parameters. In arbitrary approximation the original problem is reduced to a sequence of linear inhomogeneous boundary-value problems for plates of constant thickness. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 36–40     

On power-law fluids with the power-law index proportional to the pressure

In this short note we study special unsteady flows of a fluid whose viscosity depends on both the pressure and the shear rate. Here we consider an interesting dependence of the viscosity on the pressure and the shear rate; a power-law of the shear rate wherein the exponent depends on the pressure. The problem is important from the perspective of fluid dynamics in that we obtain solutions to a technologically relevant problem, and also from the point of view of mathematics as the analysis of the problem rests on the theory of spaces with variable exponents. We use the theory to prove the existence of solutions to generalizations of Stokes’ first and second problem.     

Extended formulations for the A-cut problem

LetG=(V, E) be an undirected graph andA⊆V. We consider the problem of finding a minimum cost set of edges whose deletion separates every pair of nodes inA. We consider two extended formulations using both node and edge variables. An edge variable formulation has previously been considered for this problem (Chopra and Rao (1991), Cunningham (1991)). We show that the LP-relaxations of the extended formulations are stronger than the LP-relaxation of the edge variable formulation (even with an extra class of valid inequalities added). This is interesting because, while the LP-relaxations of the extended formulations can be solved in polynomial time, the LP-relaxation of the edge variable formulation cannot. We also give a class of valid inequalities for one of the extended formulations. Computational results using the extended formulations are performed.     

A study of the stable equilibrium of thin shells compliant to shear and compression

Using relations of the geometrically nonlinear theory of thin shells compliant to shear and compression (the six-modal variant), we have written the key equations for determining their initial post-critical state by the finite element method. A specific feature of this model lies in the semidiscretization of the vector of displacements of an elastic body with respect to its variable thickness, based on the Timoshenko–Mindlin kinematic hypotheses, with preservation of the total vector of rotations of a normal to the median surface. We have solved numerically the problem of the stability of a circular plate, clamped over its contour, under the action of radial compressive forces, distributed uniformly along the contour. We have also performed a comparative analysis of the numerical solutions obtained and data known from the literature.     

Solution of a thermoelasticity problem for layered plates according to a more precise model

An applied theory is constructed for layered plates taking into account deformations due to transverse shear in a stationary thermal field. Equilibrium equations and boundary conditions are derived. It is shown that for plates of average thickness, consideration of transverse shear deformations and thermal expansion in the transverse direction leads to results significantly differing from those calculated according to the classical theory.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 25–29, 1989.     

Numerical solution of the problem of nonlinear deformation of an immersed cylindrical panel

The paper presents a numerical analysis of the deformation of an infinitely long flexible cylindrical panel of variable thickness immersed in a liquid with a concentrated force applied along the panel generator in the symmetry plane. The nonlinear boundary-value problem is solved by the linearization method combined with the discrete orthogonalization method. We examine the effect of variable thickness on the deformation of the panel for various initial values of the radius of curvature.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 51–54, 1988.     

The coupled thermoelastic problem for a plane with a rectilinear crack

We solve the thermoelastic problem for a plane with a rectilinear heat-conducting crack whose conductivity depends on its opening. By modeling the crack as a thin inclusion of variable thickness we reduce the problem to a system of singular integrodifferential equations for the potential densities of the temperature field. We study the behavior of the unknown functions at the ends of the contour of integration and, using a numerical-iteration method, we also determine the solution of the problem. We find an approximate asymptotic solution in the case of a weakly conducting crack.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 54–58.     

The stressed state of a thermosensitive annular plate of variable thickness

We find an exact solution of the problem of displacements and stresses in a polar-orthotropic annular plate of variable thickness with physico-mechanical properties of the material depending on temperature. The solutions are obtained as generalized and degenerate hypergeometric functions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 42–45.     

Assessment of transmission of the shear stress in potted anchors for composite rods. 2. Sleeve of variable thickness

The key problem facing the application of FRP stay cables and tendons is the anchor age. In the first part of the investigation, a model of a potted (bond-type) anchor was proposed, and a parametric analysis of the solution for the distribution of the interfacial shear stress in side the anchor with a sleeve of constant thickness was made. In this, second part, the solution is generalized to consider sleeves with a variable thickness (stepped and tapered), and the parametric analysis is extended. All the results obtained are compared, and the most effective means for decreasing the peak interfacial shear stress in the anchor are high lighted.     

Optimal assignment of NOP due-dates and sequencing in a single machine shop

We consider the problem of optimal assignment of NOP due-dates ton jobs and sequencing them on a single machine to minimize a penalty function depending on the values of assigned constant waiting allowance and maximum job tardiness. It is shown that the earliest due date (EDD) order is an optimal sequence. For finding optimal constant waiting allowance, we reduce the problem to a multiple objective piecewise linear programming with single variable. An efficient algorithm for optimal solution of the problem is given.     

Steady-state transverse vibrations of sectional cylindrical anisotropic plates of variable thickness

Using the perturbation method we solve the problem of the steady-state transverse vibrations of a plate of variable thickness consisting of circular rings made of different cylindrically anisotropic materials. Numerical studies are carried out for plates consisting of two rings with different laws of variation of thickness. We give the graphs of the distribution of values of the bending moments. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 92–95.     

Approximation of a function of two variables by a product of functions of one variable on a given domain

We are concerned with the problem of uniform approximation of a continuous function of two variables by a product of continuous functions of one variable on some domain D. This problem have been examined so far only on a rectangular domain D = U × V, where U and V are compact sets. An algorithm to give a solution of this problem in the discrete case is available. We put forward an algorithm which in certain cases allows one to construct an approximate solution of the problem on a given domain (not necessarily rectangular). This approximate solution is built in the form of interpolating natural splines, which in turn are constructed by means of discrete approximation. Depending on the degree of the splines, the problem can be solved in classes of functions with appropriate degree of smoothness.     

Solution of the stationary problem of heat conduction of laminated anisotropic inhomogeneous plates by the method of initial functions

The problem of stationary heat conduction of laminated plates of constant and variable thickness is formulated in the three-dimensional statement. We reduce the three-dimensional problem to a twodimensional one by the method of initial functions. For plates with layers of variable thickness, a system of resolving equations with variable coefficients is obtained. The obtained two-dimensional boundary-value problems are analyzed. For plates with homogeneous layers of constant thickness, we construct a solution in an analytic form. It is shown that this solution coincides with a solution obtained by the method of separation of variables.     

Optimization of arches using genetic algorithm

An optimization procedure is presented for the minimum weight and strain energy optimization for arch structures subjected to constraints on stress, displacement and weight responses. Both thickness and shape variables defining the natural line of the arch are considered. The computer program which is developed in this study can be used to optimize thick, thin and variable thickness curved beams/arches. An automated optimization procedure is adopted which integrates finite element analysis, parametric cubic spline geometry definition, automatic mesh generation and genetic algorithm methods. Several examples are presented to illustrate optimal arch structures with smooth shapes and thickness variations. The changes in the relative contributions of the bending, membrane and shear strain energies are monitored during the whole process of optimization.     

Modelling of shear localization in elastic material by means of energy relaxation

A new approach to the problem of shear localization is proposed. It is based on the energy minimization principle associated with micro–structure developments and the micro–shearing of a rank–one laminate which is aligned to shear band. The thickness of the shear band represented by the volume fraction is assumed to tend to zero. The non–convex energy due to the formation of the shear band is solved by energy relaxation in order to ensure that the problem is well–posed. An application of the proposed formulation to isotropic linear elastic material is presented. The capability of the proposed energy relaxation is demonstrated through numerical simulation of a plane strain tension. The numerical results demonstrate that there is no mesh–dependence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization of the thickness of layers of sandwich conical shells

We deal with an optimal control problem for variational inequalities, where the linear operators as well as the convex sets of possible states depend on the control parameter. The existence theorem for optimal control is applied to optimal design problems for sandwich conical shells where a variable thickness of given layers appears as a control variable.     

Optimal Control of an Axisymmetric Conical Shell with a Unilateral Elastic Foundation and Rigid Supports

We study an optimal control problem for an elastic conical shell. The displacement vector u =(u,v) is then a function of one variable s∈(a,b), 0<a<b<∞. Here u is the meridional and v the normal displacement. The shell is assumed to be simply supported. Moreover, we consider several unilateral obstacles and an unilateral elastic foundation of Winkler type i.e. the reaction force is proportional of the positive part of the normal displacement. The state problem is formulated in a form of variational inequality for u . The design parameter e =(t,z,F) includes the variable thickness of the shell, the stiffness characteristics of the foundation and the friction coefficient. The existence of the optimal control will be explained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Problem of optimal control by the natural frequency of oscillations of an orthotropic shell of revolution and its finite-dimensional approximation

The questions of optimization in problems of oscillations in orthotropic shells of revolution of variable thickness are studied for the case when the thickness and radius of curvature of the shell generatrix are used as the controls. Restrictions are imposed on the principal oscillation eigenfrequency, thickness, internal volume and other parameters. It is shown that a solution of the problem exists and, that the problem can be approximated by a sequence of the finite-dimensional problems. Certain questions of the optimal control in the problem concerning the oscillations of plates of variable thickness with the thickness serving as the control, were studied in /1–4/.