S. A. Lychev

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.

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.

Coupled dynamic thermoviscoelasticity problem

We construct a closed analytic solution of a coupled dynamic thermoviscoelasticity problem for bodies of canonical shape. The solution is represented in the form of spectral expansions in a biorthogonal eigenfunction system of the nonself-adjoint pencil of differential operators generated by the problem under study. The spectral expansions are obtained with the help of a special class of nonsymmetric integral transforms.

Nonstationary vibrations of a discretely accreted thermoelastic parallelepiped

A procedure for determining nonstationary vibrations of a discretely accreted thermoelastic body in the approximation of small deformations and thermal flows is developed. A closed-form solution is constructed for a growing parallelepiped under “smoothly rigid” heat-insulated fixation conditions for the stationary faces and the growing load-free face. The temperature field on the growing face is analyzed numerically for various accretion scenarios.

Universal deformations of growing solids

A class of universal deformations of accreted hyperelastic incompressible bodies is studied. Accretion is realized by adding prestrained layers [1–4]. The deformations correspond layerwise to the transformation of a parallelepiped to a hollow circular cylinder. Discrete and continuous accretion modes are considered and classified. Solutions of the boundary-value problems for the elastic Mooney-Rivlin potential are constructed. The solutions of the discrete accretion problems are shown to converge to solutions of the corresponding problems of continuous accretion as the number of layers increases and the layer thickness decreases.

Differential operators associated with the equations of motion and nondissipative heat conduction in the Green-Naghdi theory of thermoelasticity

We consider a wave (hyperbolic) heat conduction theory of the Green-Naghdi type. In the framework of the continual approach, such a theory permits describing low-temperature phenomena of quantum nature (for example, the second sound effect) that are not within reach of the classical parabolic thermoelasticity. At low temperatures, the medium heat conductivity experiences a substantial increase, which should be reckoned with in the design and analysis of cryogenic devices. This change in heat conduction properties results from the wave character of heat propagation, which cannot be taken into account in classical heat conductivity models of the diffusion type but can be described by hyperbolic models of the Green-Naghdi type. That is why considerable attention has been paid to the development of hyperbolic heat conduction models. The present paper deals with analytic solution methods for the corresponding nonself-adjoint initial-boundary value problems.

Various decomposition techniques for linear equations of continuum mechanics

Various methods of decomposition of a sufficiently general linear set of equations, special cases of which are frequently encountered in continuum mechanics, are described. These methods are based on splitting the sets of coupled 3D equations into several simpler independent equations. In addition to the firstorder decomposition, we also considered higher order decompositions. Examples of the decomposition of sets of equations describing slow motions of viscous and viscoelastic incompressible fluids and viscous compressible barotropic fluids and gases are presented. The obtained results can be used for the exact or numerical solution of 3D problems of continuum mechanics and physics.

Mathematical modeling of growth processes in nature and engineering: A variational approach

We present a variational approach to the mathematical theory of accreted solids. One main point in this approach is that the operator of the accretion problem proves to be self-adjoint with respect to an appropriately modified convolution bilinear form, and it is this linear form that we use in the construction of the variational functional. Our growing solid model can be efficiently applied to describe processes such as concreting, pyrolytic deposition, laser spraying, electrolytic deposition, polymerization, solidification of melts, crystal growth, glacier and ice cover freezing, sedimentary and volcanic rock forming, and biological tissue growth. These applications will be considered elsewhere.