Alexander V. Manzhirov

On a method of solving two-dimensional integral equations of axisymmetric contact problems for bodies with complex rheology

Abstract A two-dimensional integral equatin appearing in axisymraetric contact problems for bodies with complex rheology is studied. A method of constructing the solution of this equation in proposed, based on inspecting the non-classical spectral properties of an integral operator. A contact problem for a non-uniformly aging viscoelastic foundation is solved as an example.

The general non-inertial initial-boundaryvalue problem for a viscoelastic ageing solid with piecewise-continuous accretion☆

Abstract The deformation processes in a viscoelastic body whose composition, mass or volume varies in a piecewise-continuous manner due to the addition of new material to the outer surface of the body is studied. Modelling of such processes leads to basically new non-classical problems in solid mechanics. The formulation is considered and a method of constructing the solution of the general non-inertial initial-boundary-value problem for a viscoelastic ageing body with piecewise-continuous accretion is suggested. The fundamental theorems are stated and some qualitative features of the evolution of the stress-strain state of the growing body are studied.

Arch structure erection by an additive manufacturing technology under the action of the gravity force

The paper studies the influence of the gravity forces on gradually constructed objects in the field of gravity. The cases of viscoelastic aging and pure elastic materials are considered. A model of erection of a circular (semicircular) arch structure on a smooth rigid horizontal basement by the method of layered thickening of a previously mounted blank structure from inside is developed in the framework of the linear mechanics of accreted bodies. The model takes into account the influence of gravity forces on the erected structure during the entire process of its erection. In addition, the model allows creating prestresses in the accreted structure elements in the process of erection and admits arbitrary time-variable loads on the outer surface of the erected arch. The corresponding essentially two-dimensional problems of mechanics of quasistatic deformation of the considered structure before its accretion, at the stages of continuous accretion, in the pauses between these stages, and after the accretion are posed in the plane strain state approximation. The analytic solutions of these problems are constructed as series and by quadratures, which permits tracing the evolution of the structure stress-strain states at all stages listed above. These studies can be used to analyze objects produced by additive manufacturing technologies.

Closed solutions of boundary-value problems of coupled thermoelasticity

Coupled equations of thermoelasticity take into account the effect of nonuniform heating on the medium deformation and that of the dilatation rate on the temperature distribution. As a rule, the coupling coefficients are small and it is assumed, sometimes without proper justification, that the effect of the dilatation rate on the heat conduction process can be neglected. The aim of the present paper is to construct analytical solutions of some model boundary-value problems for a thermoelastic bounded body and to determine the body characteristic dimensions and the medium thermomechanical moduli forwhich it is necessary to take into account that the temperature and displacement fields are coupled. We consider some models constructed on the basis of the Fourier heat conduction law and the generalized Cattaneo-Jeffreys law in which the heat flux inertia is taken into account. The solution is constructed as an expansion in a biorthogonal system of eigenfunctions of the nonself-adjoint operator pencil generated by the coupled equations of motion and heat conduction. For the model problem, we choose a special class of boundary conditions that allows us to exactly determine the pencil eigenvalues.

Influence of the erection regime on the stress state of a viscoelastic arched structure erected by an additive technology under the force of gravity

We study the influence of gravity forces on additively constructed objects of a viscoelastic aging material (in a special case, of a purely elastic material) in the absence of additional surface loads and prestresses in the accreted material elements. It is shown that the stress-strain state of such objects crucially depends on how the process of their gradual formation evolves in time. The main tendencies whose interaction determines the process of deformation of these objects under a given formation regime are revealed and analyzed. The general reasoning is illustrated by the results of numerous numerical experiments performed in the framework of the model of linear mechanics of accreted bodies, which was developed by the authors for studying the essentially two-dimensional engineering problem on the erection of a heavy circular arched structure (a semicircular vault) on a smooth horizontal base by the method of layer-by-layer thickening of a blank structure previously erected on the base. This problem is used as an example in the detailed studies of the influence of the erection regime of a viscoelastic aging structure on the development of its stress state. We show that it is very important to take into account the influence of gravity forces during the entire process of erection of heavy objects rather than in their final configuration. It is conclusively shown that, without considering this influence, one can arrive at completely false conclusions about the current and resulting states of the erected structures such as overestimation of their strength and stability at the stage of formation and of their bearing capacity in their operation. The possibilities of efficient control of the stress state of the considered arch structure by varying the rate of the additional material accretion to the structure are demonstrated.

Mechanics of Growing Solids and Phase Transitions

Phase transitions can be usually observed in nature and technology which effectively utilize certain types of these transitions. An approach to modeling phase transition processes on the basis of the mathematical theory of growing solids is developed. Liquid-solid and gas-solid phase transitions are under consideration. Main attention is paid to the processes of solid phase growth and deformation.

A mixed integral equation of mechanics and a generalized projection method of its solution

A mixed multidimensional integral equation containing integral operators of various types is studied. The case in which the equation has one compact, self-adjoint, and strongly positive operator (with constant limits of integration) and two non-self-adjoint integral Volterra operators (with a variable upper limit of integration) is considered. To solve the equation, an effective projection method allowing one to obtain the result in a form with explicitly distinguished principal singularities is proposed.

Mechanics of Growing Solids: New Track in Mechanical Engineering

A vast majority of objects around us arise from some growth processes. Many natural phenomena such as growth of biological tissues, glaciers, blocks of sedimentary and volcanic rocks, and space objects may serve as examples. Similar processes determine specific features of many industrial processes which include crystal growth, laser deposition, melt solidification, electrolytic formation, pyrolytic deposition, polymerization and concreting technologies. Recent researches indicates that growing solids exhibit properties dramatically different from those of conventional solids, and the classical solid mechanics cannot be used to model their behavior. The old approaches should be replaced by new ideas and methods of modern mechanics, mathematics, physics, and engineering sciences. Thus, there is a new track in solid mechanic that deals with the construction of adequate models for solid growth processes. The fundamentals of the mathematical theory of growing solids are under consideration. We focus on the surface growth when deposition of a new material occurs at the boundary of a growing solid. Two approaches are discussed. The first one deals with the direct formulation of the mathematical theory of continuous growth in the case of small deformations. The second one is designed for the solution of nonlinear problems in the case of finite deformations. It is based on the ideas of the theory of inhomogeneous solids and regards continuous growth as the limit case of discrete growth. The constitutive equations and boundary conditions for growing solids are presented. Non-classical boundary value problems are formulated. Methods for solving these problems are proposed.Copyright

Reference configurations of growing bodies

The growing bodies are considered as bodies with induced inhomogeneity caused by junction of inconsistently deformed parts. The body is formalized as an abstract smooth manifold, and all possible affine connections on it are classified. A method is shown for introducing a special connection—material connection—for which neighborhoods of all material points of a growing body pass into a stress-free state. A method for constructing a global stress-free reference configuration of a growing body as an embedding in a space with absolute parallelism is proposed. It is shown that in the case of layered accretion, the material connection corresponding to the stress-free embedding is determined by three independent functions and is in general non-Euclidean. The property of being non-Euclidean is determined by the fact that the torsion of the material connection is nonzero. We suggest to formalize the growing body as a fibration of a three-dimensional smooth manifold over a one-dimensional base, and this formalization characterizes the structure of the material connection.

Unsteady vibration of a growing circular plate

Forced vibration of a thickening circular plate is studied within the framework of the small deformation theory. The plate material is assumed to be elastic and isotropic and the plate thickness to be continually increasing due to influx of material from outside. It is also assumed that the plate thickness varies with time but is independent of the space coordinates. Moreover, in the process of growth, the midsurface position does not change, which suggests that the plate grows symmetrically on both its surfaces.