Yu. N. Nemish

Development of analytical methods in three-dimensional problems of the statics of anisotropic bodies (Review)

The main results of scientific research carried out at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine in the field of three-dimensional problems of the statics of anisotropic bodies are stated in a systematic form. The results include the structural method of constructing the exact analytical solutions of equations of the elastic and thermoelastic equilibrium of rectilinearly orthotropic bodies and approximate analytical methods of solving three-dimensional boundary-value problems for curvilinearly orthotropic bodies of canonical and noncanonical form. Results of solution of specific boundary-value problems for orthotropic and transversally isotropic bodies are analyzed.

Stress-strain state of non-thin plates and shells. Generalized theory (survey)

ConclusionsThus, this part of our survey has presented the main approaches that have been taken to the construction of two-dimensional (in terms of the space coordinates) equations of a generalized theory of plates and shells. The solutions of these equations represent a certain approximation of the solution of the initial three-dimensional problem. They are based on expansion of the sought functions into Fourier series in Legendre polynomials of the thickness coordinate. Studies completed on the basis of the given variants of plate and shell theory were systematized and analyzed. In terms of the method of its construction, the theory involves a regular process of replacing the solution of the three-dimensional problem by the solution (or sequence of solutions) of two-dimensional boundary-value problems or initial-boundary-value problems. Numerical results illustrating the convergence of the successive approximation were presented. It should be noted that to make comparison with the results of classical or applied theories, several of the studies cited here presented solutions of problems for thin plates and shells with allowance only for the initial terms of expansions of the stress and displacement components into base functions (Legendre polynomials).

Stress-deformation state of thick shells and plates. Three-dimensional theory (review)

ConclusionsIn the first part of the present review we surveyed systematically published results of investigations of the stress-deformation state of thick-walled spheres, ellipsoids, cones, circular cylinders, as well as thick slabs, obtained by exact analytic solutions of spatial problems of elasticity theory. Several quantitative results were given of the variation of displacements and stresses with shell or plate thickness, and their comparative analysis was provided, making it possible to establish the validity limits of the corresponding applied theories. We also surveyed systematically published specific results of the spatial stress-deformation state of nearly canonical thick-walled shells, as well as non-thin plates of varying thickness, obtained by effective approximate analytic methods and known exact solutions for the corresponding canonical regions. Especially noted were characteristic mechanical (including boundary) effects on the stress-deformation state of the bodies under consideration. These effects are generated, in particular, by variations in the radius of curvature of the surface, the thickness parameter, the amplitude and frequency of the corrugated surface, material, inhomogeneity, conditions of mechanical contact between layers, the nature of self-balancing loads, and other factors.However, the possibilities of exact and effective approximate analytic solution of boundary value problems of this class in the three dimensional statement are restricted. In the case of shells and plates of mean thickness these results can be substantially supplemented by qualitative and quantitative data, obtained on the basis of analytic solutions in the generalized theory of shells and plates, based on expansions of components of the stress-deformation state in Legendre polynomial series, and making it possible, in principle, to approximate the three-dimensional solution with any required accuracy. This is one of the basic features distinguishing it from the classical and applied theories of shells and plates. The second part of this review will be devoted to systematic and comparative analysis of the results of investigations carried out within the generalized theory of non-thin shells and plates.

Determination of the stressed-strained state of nonthin transversally isotropic spherical shells with elastically stiffened curvilinear holes

An approximate analytical method has been developed for determining the stressed-strained state of nonthin transversally isotropic shallow spherical shells with noncircular elastic inclusions or stiffened holes, under both ideal and nonideal contact at the interface. The method is based on two tried and tested analytical methods—the method of Fourier expansion of the required functions using Legendre polynomials and the second version of the perturbation method for the boundary shape.

Three-dimensional problems of mechanics of deformable bodies of noncanonical shape

The paper is based on the authors report at the General Jubilee Meeting of the Mechanics Division on the occasion of the 80th anniversary of the National Academy of Sciences of Ukraine. Results obtained in the subject area “mechanics of deformable bodies of noncanonical shape” are discussed. This subject area was formed at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine on the basis of variants of the analytical method of boundary-shape perturbation proposed and developed at the Institute. The objects of investigation and the classification of three-dimensional boundary-value problems for noncanonical areas are analyzed. Tests of the accuracy of approximate solutions obtained using the developed analytical methods are indicated.

One class of exact analytical solutions of three-dimensional thermoelastic-equilibrium equations for orthotropic plates

A structural method is proposed to construct one class of analytical solutions of three-dimensional thermoelastic-equilibrium equations for rectilinearly orthotropic plates. This method allows us to establish the analytical structure of a partial solution of the inhomogeneous equations of thermoelastic equilibrium for orthtropic plates based on the known analytical structure of a temeprature field (found by solving the corresponding boundary-value problem of the stationary theory of thermoelasticity). The well-known solution of inhomogeneous equations of thermoelastic equilibrium for transversely isotropic plates follows from the obtained exact solution as a partial case. The exact general solutions of the three-dimensional homogeneous equations of elastic equilibrium are also presented. Their analytical structure is similar to the constructed partial solution corresponding to the known temperature field.

The stress state of a laminated nonlinearly elastic thick-walled spherical shell

This paper deals with spatial axisymmetric boundary-value problems of the physically nonlinear theory of elasticity for piecewise-homogeneous spherical bodies. The passage to dimensionless characteristics of the stress-strain state allows us to extract a physical dimensionless small parameter in the nonlinear state equations. The solution of nonlinear equilibrium equations and boundary-value problems is searched for in the form of series in positive degrees of the small parameter. This approach allows reducing the stated physically nonlinear boundary-value problem to a sequence of corresponding linear nonhomogeneous problems. A specific analytical solution and numerical results are obtained for a two-layer nonlinearly elastic spherical shell under bilateral pressure.

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.

Solution of three-dimensional boundary-value problems for laminated noncircular cylinders

UDC 539.3 A second variant of the boundary perturbation method was developed in the monograph [4] to solve three-dimensional boundary-value problems of the mechanics of deformable bodies bounded by nonorthogonal (in particular, noncircular cylindrical) surfaces (surfaces for which conditions of orthogonality are not satisfied at arbitrary points between the unit normal and the unit coordinate vectors). The method describes nonorthogonal surfaces by linear equations in a small parameter characterizing the amplitude of the deviation of the given surface from circular cylindrical shape. Numerous problems solved by the method and the corresponding numerical results were systematized and analyzed in [2, 4]. Although the assumptions made regarding the linearity of the equation of the interface relative to a small parameter simplified the mathematical calculations, it limited the generality of the approach, the range of problems that can be solved, and their practical value. Some of these deficiencies are corrected in the present study. Here, we generalize the second variant of the boundary-perturbation method to the case when the noncircular cylindrical interfaces of a laminated body are described by nonlinear parametric equations.

The S. P. Timoshenko institute of mechanics at the beginning of the third millennium

A brief historical review of the main scientific activities at the S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, is outlined. Current trends are examined, in the Institute as a whole and in particular departments. As a resource, reviews of the Institutes activity published between 1955 and 1999 in this journal and elsewhere are listed in a bibliography.