No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Contact problems of the mechanics of bodies with accretion
Abstract
Contact problems of the mechanics of bodies with accretion are studied. A general formulation of the mixed problem is given for a viscoelastic ageing body during its continuous piecewise accretion. Complete systems of equations of the mixed problem are given in time intervals from the onset of loading to the onset of accretion, from the onset of accretion to the end of accretion, and beyond it.The characteristic feature of the basic relations in the case of a body with continuous accretion is the use not of the usual equations of compatibility of the deformations and the Cauchy relations, but of their analogues in the rates of change of the corresponding quantities /1–3/. Moreover, the given previous histories of the deformation tensor of the accruing elements form, at the instant of attachment, specific initial and boundary conditions /2/ on the accruing surface. In particular, the total stress tensor associated with external loads and characterizing the tightness of attachment of the accruing elements is determined at the accruing surface /2, 3/. The instant of attachment of the new elements to the main body represents an important characteristic of the process. The set of instants of attachment completely determines the configuration of the accruing body at any instant of time. Equations of state of the theory of creep of the inhomogeneously ageing bodies are used /4, 5/. The equations reflect the fundamental specific features of the accretion process where the times of preparation and onset of loading play an important part.A method of solving the mixed and initial-boundary value problems is given. Contact problems for a wedge under various methods of accretion are considered. Integral equations are derived and their solutions constructed. Numerical solutions of the contact problems for a wedge with accretion are given for the case when the influx of matter from outside results in increasing the wedge angle, and for an accruing quarter-plane. Qualitative and quantitative effects are discussed, especially the influence of the method and rate of accretion on the contact characteristics.
The paper deals with various statements and mathematical models of contact and contact‐wear problems for bodies with coatings. It is shown that the mathematical models for a number of such problems can be represented as a mixed integral equation or a system of mixed integral equations with additional conditions. It is also shown that these equations contain rapidly varying or even discontinuous functions in the case of interaction between bodies of complex shape and with some surface properties. Therefore, it is necessary to use a special approach for constructing efficient analytic solutions. Its implementation is demonstrated by an example.
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.
The contact problem of the torsion of a viscoelastic ageing, growing cylinder by a rigid stamp is considered. Dual series equations reflecting the mathematical content of the problem of different stages of the growing process are derived and investigated. The results of a numerical analysis and the singularities of the qualitative behaviour of the fundamental characteristics are discussed.
A general theory is proposed for the analysis of small deformations in growing bodies. The growing nature of these bodies is due to reasons that are not related to the main deformation process, as for example is the case in solidifying or melting bodies and in magnetic tapes during winding. We assume that the growing boundary is completely described in time. The strain tensor in such growing bodies does not satisfy the compatibility conditions. However, the strain rate tensor satisfies the compatibility condition and as such a hypoelastic constitutive model is proposed as an appropriate elastic model and the whole deformation problem is casted in the form of an initial/boundary value problem. The obtained solution satisfies continuity of deformation and it can easily be extended to account for rate dependent phenomena.
As an application of this theory, a two-dimensional axially symmetric model and its finite element implementation for computing the induced stresses and deformation in a magnetic tape pack during a winding operation, are presented. The effects on the stresses of a tape pack due to a non-uniform across the tape width applied winding tension, are examined and presented.
The problem of the impression of a stamp with arbitrary base in the face of an elastic wedge with the other face free is studied by an extension of the “small λ” asymptotic method /1/. Solutions are found specifically for an inclined and parabolic stamp. To confirm the accuracy of the asymptotic solutions, the problem is also investigated by a reduction to two infinite algebraic systems of equations of the second kind. The results can be utilized to refine the procedure for computing the contact strength of gear transmission. Only the case of a stamp with a flat base, which is of the least interest for applications, has been studied previously /2, 3/.
Plane contact problems of the interaction between inhomogeneous ageing viscoelastic foundations and arbitrary finite systems of rigid stamps, not applied and removed simultaneously, are investigated. Formulations of the problems are given. Systems of resolving two-dimensional integral equations are derived and methods are proposed for their solution. Numerical computations are presented for different kinds of ageing during interaction between a concrete foundation and two dissimilar stamps not applied simultaneously. Qualitative effects are discussed.
Papers [1,2] are used as the basis for the formulation of a boundary value problem of the theory of creep for a nonuniformly aging body, with the elements of varying ages accumulating on the body in a continuous or discrete manner. The initial equations are given and conditions formulated which determine the solution of the boundary value problem of the theory of creep for such bodies. The characteristic feature of these bodies is, that in the course of accumulation not only their form, surface and volume forces and the boundary conditions change, but also their physical and mechanical properties with respect to time and the coordimates.This is due to the fact that the aging process in these bodies does not follow the same course in all their elements. Such phenomena take place during a consecutive erection and loading of engineering structures, in the crystal growing processes, during the phase transitions in viscoelastic bodies, etc.The basic papers dealing with solutions of the problems of accretion using the methods of the theory of elasticity are given in [3], The theory of elastic body with creep is used in [4] to study the stress-strain state in homogeneous bodies with accretion. A more general formulation is used for the same problem in [5].
Boundary value problem of the theory of creep for a body with accretion Non-linear problems of the theory of creep of accruing bodies, subjected to ageing
- N Kb
- N Arijtyunyan
- Metlov Kb
REFERENCES ARUTYUNYAN N.KB., Boundary value problem of the theory of creep for a body with accretion. PMM, 41, 5, 1977. ARIJTYUNYAN N.KB. and METLOV V.V., Non-linear problems of the theory of creep of accruing bodies, subjected to ageing. Izv. Akad. Nauk SSSR, MTT, 4, 1983. ARUTYUNYAN N.KIi., DROZDOV A.D. and NAUMOV V.E., Mechanics of Growing Viscoelastic Bodies.
Some problems of thetheoryof creep for inhomogeneously ageing bodies
- Nauka
- Moscow
Nauka, Moscow, 1987. 4. ARUTYUNYAN N.KH., Some problems of thetheoryof creep for inhomogeneously ageing bodies. Izv. Akad. Nauk SSSR, MTT, 3, 1976. 5. ARYTYUNYAN N.KH. and KOLMANOVSKII V.B., Theory of Creep of Inhomogeneous Bodies. Nauka, Moscow, 1983.
Integral Transforms in Problems of the Theory of Elasticity
- Uflyand S Ya
UFLYAND YA.S., Integral Transforms in Problems of the Theory of Elasticity. Nauka, Moscow, Leningrad, 1967. 9. MARKUSHEVICH A.I., A Short Course in the Theory of Analytic Functions.
Contact Problems for an Elastic Wedge
- Nauka
- V M Moscow
Nauka, Moscow, 1978. 10. ALEKSANDROV V.M., Contact Problems for an Elastic Wedge. Inzh. Zh. MTT, 2, 1967.
Tables of Integrals, Sums, Series and Products. Nauka, Moscow, 1971. 13. SHTAYERMAN I.YA., The Contact Problem of the Theory of Elasticity
- Nauka
- Moscow
- I S Gradshtein
Nauka, Moscow, 1974. 12. GRADSHTEIN I.S. and RYZHIK I.M., Tables of Integrals, Sums, Series and Products. Nauka, Moscow, 1971. 13. SHTAYERMAN I.YA., The Contact Problem of the Theory of Elasticity. Gostekhizdat, Moscow, Leningrad, 1949. 14. ARUTYUNYAN N.KH., Some Problems of the Theory of Creep. Gostekhizdat, Moscow, Leningrad, 1952.
Non-linear problems of the theory of creep of accruing bodies, subjected to ageing
- Arutyunyan
Some problems of the theory of creep for inhomogeneously ageing bodies
- Arutyunyan
Boundary value problem of the theory of creep for a body with accretion
- Arutyunyan
Fundamental solutions of the problems for a growing body in the form of a quarter-plane
- Arutyunyan