Fractional boundary value problems with singularities in space variables

We are concerned with the existence of solutions for the singular fractional boundary value problem $^{c}\kern-1pt D^{\alpha}u = f(t,u)$ , u(0)+u(1)=0, u′(0)=0, where α∈(1,2), f∈C([0,1]×(??{0})) and lim _x→0 f(t,x)=∞ for all t∈[0,1]. Here, $^{c}\kern-1pt D$ is the Caputo fractional derivative. Increasing solutions of the problem vanish at points of (0,1), that is, they “pass through” the singularity of f inside of (0,1). The results are based on combining regularization and sequential techniques with a nonlinear alternative. In limit processes, the Vitali convergence theorem is used.     

On finite-element solution of spatial thermoviscoelastoplastic problems

A finite-element algorithm is proposed for the analysis of the thermoviscoelastoplastic stress-strain state of bodies under complex loading (thermal and mechanical). It is assumed that an arbitrary element of the body deforms along a rectilinear or slightly curved path. The three-dimensional stress-strain state of the body’s elements is determined using the iterative method of additional strains. The technique is tested by analyzing the three-dimensional viscoelastic stress-strain state of a hollow cylinder and the thermoplastic state of a disk __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 16–25, May 2006.     

On the solution of three-dimensional boundary-value problems for layered bodies with nonplanar interfaces

The interfaces play an important role in various buildup bodies, and also in the composite materials and structural elements. Special monographs [7, 8] have been devoted to this question, presenting the results of scientific studies of the physical and chemical phenomena on the interfaces, the mechanical behavior, and the role of the interfaces in the damage processes, and also their influence on the basic mechanical properties of the composites. In many cases the interfaces deviate from the ideal geometric shapes: planar (in the layered composites), circular cylindrical (in the fibrous composites), and spherical (in the granular composites). Numerous theoretical and experimental studies confirm this. Thus, in the explosive welding of metals (and nonmetals) there form wavy surfaces, the sections of which may be close to sinusoids, for example in the welding of niobium and copper [9]. If the densities of the materials differ significantly, then the sinusoidal nature of the interface distorts as illustrated in [12] for the example of the welding of lead and steel. In addition, in view of the nature of the technological processes [10] the interfaces may become curved in the layered composite materials and deviate locally or periodically from the ideal coordinate planes. Theoretical and experimental studies have shown that the shape of the interface has a significant influence on the physical and mechanical processes and phenomena (bond strength, stress concentration, wave diffraction, thermal conduction, and so on). Numerous publications that are cited in the survey works [1, 3, 11] confirm this. A second variant of the boundary shape perturbation method was developed in [4, 5] for the solution of the three-dimensional boundary-value problems for nonorthogonal surfaces that are close to the coordinate planes. It was assumed that the equations of the interfaces are linear relative to the small parameter characterizing the degree of deviation from the coordinate planes. This narrowed significantly the class of the examined boundary-value problems and their practical importance. In the present work we examine the three-dimensional boundary-value problems of the mechanics of layered bodies with interfaces that are described by nonlinear equations relative to a small parameter. We construct in general form the recurrence relations and the differential operators of the boundary conditions, making it possible to solve the three-dimensional boundary-value problems with the accuracy that is required for applications. We examine particular cases and present one of the possible criteria for evaluating the accuracy of the approximate solutions that are obtained with the aid of the described variant of the boundary shape perturbation method.S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 23–32, February, 1994.     

Complementary energy variational approach for plane elastic problems with singularities

Presented is the numerical analysis of plane elastic problems involving stress concentrations and/or singularities using a physically meaningful complementary energy variational approach. The continuum body is modeled by a non-conventional truss structure. Stress distributions in laminated composite bodies and orthotropic sheets with a through crack are obtained. The present results are compared with the analytical solutions for different numerical methods.     

On the numerical solution of some two-dimensional boundary-contact delocalization problems

The elastic equilibrium of a multi-layer confocal elliptic ring is studied. The ring consists of steel, rubber and celluloid layers which differ in thickness and in the order in which they are placed relative to one another. Using the solutions of the considered problems, the following delocalization problem is solved: for a three-layer elliptic body, the external elliptic boundary of which is loaded by normal point force and the internal boundary is stress-free and the layers of which are in rigid or sliding contact with one another, by an appropriate choice of layer thickness and arrangement of the layers relative to one another we can obtain a sufficiently uniform distribution of normal displacements on the internal elliptic boundary. Numerical solutions are obtained by the boundary element method and the related graphs are constructed. For the two-layer ellipse, exact and approximate solutions of the same problem are obtained respectively by the method of separation of variables and by the boundary element method. The results obtained by both methods are compared and the conclusion as to the reliability of the numerical boundary element method is made.     

On singularities associated with the curvilinear anisotropic elastic symmetries

The objective of this paper is to describe a different approach to modeling the material symmetry associated with singularities that can occur in curvilinear anisotropic elastic symmetries. In this analysis, the intrinsic non-linearity of a cylindrically anisotropic problem is demonstrated. We prove that a simple homogenization process applied to a representative volume element containing the cylindrical anisotropic singularity removes the singularity. This geometric and interpretive approach is an aid to better modeling of material symmetry associated with these singularities.     

On the solution of certain field problems by the schwarz-christoffel method

In many practical cases the usefulness of the Schwarz-Christoffel method to solve two-dimensional field problems (Laplace equation with Dirichlet boundary conditions) is limited by the presence of transcendental functions of complex variables. We demonstrate here a new technique whereby, in lieu of qualitative plots of equipotential surfaces and flux lines, field components and potential can be expressed as real power series of the coordinates (x, y). The convergence of these series is only limited by the proximity of singular points corresponding to the physical convex corners. By choosing suitable points on the boundary around which the series of expansion are developed, fringing field components in the regions of interest between the boundaries can be computed directly. In some cases the series converges rapidly and assumes a remarkably simple form.     

On approximate solution of boundary-value problems by the least squares method

Using the least squares method, we construct a new iterative procedure for finding solutions of a weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case in the form of an expansion of a solution in a generalized Fourier polynomial in the neighborhood of the generating solution. We obtain an estimate for the range of values of the small parameter for which this iterative procedure converges to the required solution. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 554–573, October–December, 2008.     

On the displacement potential solution of plane problems of structural mechanics with mixed boundary conditions

The present paper describes the advancement of displacement potential approach in relation to solution of plane problems of structural mechanics with mixed mode of boundary conditions. Both the conditions of the plane stress and the plane strain are considered for analyzing the displacement and stress fields of the structural problem. Using the finite difference technique based on the present displacement potential approach for the case of the plane stress and the plane strain conditions, firstly an elastic cantilever beam subjected to a pure shear at its tip is solved and these two solutions (plane stress and plane strain) are compared with Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951); secondly the above-mentioned displacement potential approach for the case of the plane stress and the plane strain conditions are applied to solve a one-end fixed square plate subjected to a combined loading at its tip. Effects of plane stress and plane strain on the elastic field of the plate are discussed in a comparative fashion. Limitations of Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951) over the displacement potential approach for the case of the plane stress and the plane strain conditions are not only discussed but also the superiority of the present displacement potential approach for the case of the plane stress and the plane strain conditions are reflected in the present research work.     

On the approximate solution of autonomous boundary-value problems by the least-squares method

Using the least-squares method, we construct a new iterative procedure for finding solutions of an autonomous weakly nonlinear boundary-value problem in the critical case in the form of a generalized Fourier polynomial expansion.     

On the general equation and the general solution in problems for plastodynamics with rigid-plastic material

This work is the continuation of the discussion of refs. [1–2]. We discuss the dynamics problems of ideal rigid — plastic material in the flow theory of plasticity in this paper. From introduction of the theory of functions of complex variable under Dirac-Pauli representation we can obtain a group of the so-called general equations (i.e. have two scalar equations) expressed by the stream function and the theoretical ratio. In this paper we also testify that the equation of evolution for time in plastodynamics problems is neither dissipative nor disperive, and the eigen-equation in plastodynamics problems is a stationary Schrödinger equation, in which we take partial tensor of stress-increment as eigenfunctions and take theoretical ratio as eigenvalues. Thus, we turn nonlinear plastodynamics problems into the solution of linear stationary Schrödinger equation, and from this we can obtain the general solution of plastodynamics problems with rigid-plastic material.     

On the use of the complementary energy in the solution of buckling problems

A systematic derivation of the expression for the complementary energy in elastic buckling problems is presented. Compatibility is identified with variation with respect to the stress components, and the resulting eigenvalue problem is shown to be equivalent to, and sometimes more convenient than, the corresponding formulation in terms of the potential energy. Similarly, approximate techniques may lead to better as well as simpler estimates, whose upper bound property can, however, be assured only through appropriate safeguards.The method is applied in some detail to buckling of columns of arbitrary boundary conditions and axial force distribution. Another example is the problem of lateral beam buckling, with the effect of warping restraint included. In both cases (and presumably in many others) the complementary energy formulation is of lower order than the conventional potential energy formulation, and it is clear that the same simplification should also apply to finite elements or other discrete formats. The method is restricted to the (technically significant) case of a linear prebuckling state.     

On numerical solution of dynamic problems in the theory of reinforced shells

Dynamic problems for cylindrical shells reinforced with discrete ribs are examined. A numerical algorithm based on Richardson extrapolation is developed. Specific problems are solved, and the results are analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 50–56, May 2006.     

On the solution of plane boundary-value problems of elasticity in a rectangle

An algorithm for solving plane boundary-value problems of elasticity for a rectangular domain is expounded. The algorithm is based on a complex-valued representation of the general solution to the differential equations of the plane problem and on the use of Lagrange polynomials to satisfy the boundary conditions. The algorithm can quite easily be implemented in a computer program. This is probably the simplest way of solving boundary-value problems of this class __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 97–102, January 2006.