


Vol 229, No 2 (2018)
- Year: 2018
- Articles: 7
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14888
Article
Selected Problems of the Mechanics of Coupled Fields
Abstract
We present a brief survey of investigations carried out in recent years by the Lviv scientific school of the mechanics of coupled fields on the problems of construction and generalization of mathematical models and methods aimed at the description, determination, and optimization of the thermomechanical behavior of bodies under the combined action of force, thermal, and electromagnetic loads.



Analysis of Nonclassical Fracture Problems for Prestressed Bodies with Interacting Cracks
Abstract
We consider two types of nonclassical fracture mechanisms, namely, the fracture of cracked bodies with initial (residual) stresses acting along the crack planes and the fracture of solids under compression along parallel cracks. To investigate nonaxisymmetric and axisymmetric problems for infinite solids containing two parallel coaxial cracks or a periodic set of coaxial parallel cracks, we use a combined analytic-numerical method within the framework of the three-dimensional linearized mechanics of solids. The analysis involves the representation of stresses and displacements in the linearized theory via harmonic potential functions. With the use of the integral Fourier–Hankel transformations, the problems are reduced to the solution of Fredholm integral equations of the second kind. This approach allows us to investigate problems in a unified general form for compressible and noncompressible homogeneous isotropic or transversely isotropic elastic bodies with an arbitrary structure of the elastic potential, and the material specification of the model is carried out only in the stage of numerical analysis of the resolving equations obtained in the general form. The effects of initial stresses on the stress intensity factors are analyzed for highly elastic materials and layered composites (modeled as transversely isotropic elastic bodies). The “resonance-like” effects are revealed when compressive initial stresses reach the values corresponding to the local loss of stability of the material in the vicinity of cracks, which, according to the indicated combined method, allows one to determine the critical (limiting) load parameters under the conditions of compression of the body along the cracks. The conclusions concerning the dependences of the stress intensity factors and critical (limiting) parameters of compression on the geometric parameters of the problems are formulated as well as concerning the dependences on physical and mechanical characteristics of the materials.






Systems of Functions Orthogonal Over the Domain and Their Application in Boundary-Value Problems of Mathematical Physics
Abstract
We formulate a boundary-value problem for eigenvalues and eigenfunctions of the Helmholtz equation in a complex domain with the use of mutually conjugated complex variables. The obtained systems of functions are orthogonal in this domain and constructed by using the Bessel functions and powers of the conformal mappings of the analyzed domains onto a circle. The solutions of boundary-value problems for the principal equations of mathematical physics (hyperbolic, parabolic and elliptic types) are obtained in the form of the sums of series in the systems of functions orthogonal over the domain.



Restoration of the Initial Data in the Problem for a Diffusion Equation with Fractional Derivative with Respect to Time
Abstract
We prove the correctness of the inverse problem of finding a pair of functions: the classical solution u(x,t) of the first boundary-value problem for a linear diffusion equation with regularized fractional derivative of order α ∈ (1, 2) with respect to time in a rectangular domain (0, ℓ) × (0, t0] and unknown initial values of the function u(x,t) for the case of additionally given values of the function at a certain fixed time t0 .



Multidimensional Analogs of the Cauchy–Riemann System and Representations of Their Solutions via Harmonic Functions
Abstract
At present, there are numerous multidimensional generalizations of holomorphic vectors. The most general of these is the four-dimensional generalization of the Cauchy–Riemann system. In the present work, by introducing two quaternion functions and the notion of quaternion differentiation, we obtain, for the first time, a five-dimensional generalization of holomorphic vectors. By using the representation of holomorphic vectors via the quaternion harmonic function and its derivatives, we consider the Riemann–Hilbert problem and one problem in a layer. A new solution of the Riemann–Hilbert problem in the five-dimensional half space is obtained.



Coupled Thermoelasticity Problem for Multilayer Composite Shells of Revolution. I. Theoretical Aspects of the Problem
Abstract
On the basis of a general integral form of the variational principle of the least possible dissipation of energy of the nonequilibrium thermodynamics, we deduce a nonclassical nonsteady heat-conduction equation for multilayer polyreinforced shells of any shape. A method for the determination of the integral heat conductivities of reinforced layers is developed and the effective constitutive equations for the description of its thermoelastic behavior are proposed. A nonclassical model of deformation of the multilayer shell and a nonlinear model of distribution of the heat flux along the thickness of the layer are formulated. This allows us to take into account the transverse shear strains and guarantee the conditions of thermomechanical contact of the layers and the conditions of thermomechanical loading on the face surfaces of the shell. We construct a closed system of differential equations with the corresponding initial and boundary conditions for a coupled problem of thermoelastic deformation of layered composite shells and plates.


