Issues related to the formulation of the variational principles of generalized thermoelasticity based on the Green–Naghdi (GN-thermoelasticity or GN theory) theory are investigated. Variants of the Green–Naghdi theory are considered: classical coupled thermoelasticity and hyperbolic thermoelasticity. To construct functionals of the linear theory of thermoelasticity, the variational principle, which is a generalization of Hamilton’s principle in the linear theory of elasticity, is used. Conditions for the stationarity of the constructed functionals are formulated leading to differential formulations of thermoelasticity problems.

The work is devoted to investigating the influence of the mechanical field on temperature and diffusion processes occurring under steady bending of slender beams. The model used here takes account of the finite velocity of propagation of thermal and diffusion disturbances . A mathematical formulation of the problem includes the system of equations of unsteady flexural vibrations of the beam with account of heat and mass transfer, that has been obtained from the general thermomechanodiffusion model for continuous media using the generalized principle of virtual displacements. With the example of a hingedly supported three-component beam fabricated from a zinc, copper, and aluminum alloy and exposed to unsteady bending moments, the authors have investigated the interaction of mechanical, temperature, and diffusion fields, and also have analyzed the influence of relaxation effects on the kinetics of heat and mass transfer.

In the paper, the coupled elastic-diffusion processes arising as a result of unsteady bending vibrations of an orthotropic plate that has a cantilever fastening on one side and hinged support on the sides adjacent to the cantilever have been analyzed. For a mathematical description of physical and mechanical processes, the Timoshenko plate model supplemented with mass transfer equations taking into account the finite speed of propagation of diffusion flows was used. The solution algorithm was based on the use of the equivalent boundary conditions method allowing to express the solution to the problem posed through a known solution to some auxiliary problem of a given class. The nature of the interaction between mechanical and diffusion fields was simulated using the example of a bendable three-component plate.

The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier’s separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions.

The effect of a heat flux of constant intensity on a circular three-layer plate, thermally insulated along the contour and lower surface, is considered. A solution of the problem of thermal conductivity of the plate with thickness-averaged thermophysical parameters of materials is presented. The nonstationary temperature field is nonuniform over the thickness of the plate. It is shown that on instantaneous drop, the heat flux can cause sagging and free vibrations of the three-layer plate. The kinematics of the plate package obeys the broken line hypothesis. After applying a load, the normal in thin load-bearing layers does not change its length and remains perpendicular to the middle surface of the layer. In a relatively thick filler, the deformed normal retains its length and straightness, but is rotated by a small additional angle, i.e., the shift is taken into account. The formulation of the corresponding initial boundary-value problem is given. The equations of motion were obtained using the variational method with account for the transverse forces of inertia. The boundary pivot conditions are accepted on the contour of the plate. Radial movements in the layers are expressed through three sought functions: plate sagging, shear, and radial movement of the middle plane of the filler. It is shown that these sought functions satisfy the inhomogeneous system of three differential equations. To solve the system, the method of series expansion in the constructed fundamental system of eigenorthonormal functions was used. A transcendental equation is written out to obtain the corresponding eigenvalues. A numerical parametric analysis of the solution was carried out depending on the geometric and thermophysical characteristics of the layer materials and the time of exposure to the heat flux.

Non-stationary longitudinal vibrations of a moment elastic rod of finite length are investigated. To describe the motion of the rod, the system of equations of the general model of moment elastic thin bodies is used without additional hypotheses. The equations of this model take into account longitudinal movements, changes in the angle of independent microrotation, as well as transverse compression of the rod. The rod material is assumed to be homogeneous and isotropic. The system of equations of motion is supplemented by physical relations that describe the relationship of displacements, changes in angles and transverse compression with forces. In contrast to classical models, in addition to normal forces, additional force factors arise in a moment rod. They are: additional moments, moment cutting forces, moments of moment stresses. Accordingly, in addition to the elastic constants of the material, additional physical parameters of the medium are taken into account, which are necessary when taking into account moment effects in the material. The conditions of generalized hinged support are used as boundary conditions at the ends of the rod. The initial conditions are assumed to be zero. To construct the solution, expansions of the desired functions and the external load into trigonometric Fourier series are used. Substituting these expansions into the original relations leads to a system of equations for the coefficients of time-dependent series. To solve it, the integral Laplace transform a in time is used. As a result, expressions for the required coefficients of expansion series in the image space are found. Each of these expressions is the sum of three products. The factors in these products are the Laplace images of the coefficients of the Fourier expansions for the load and for the influence functions. Influence functions are fundamental solutions (Green's functions) of the problem under study. The original coefficients of the series for the influence functions are found analytically using residues. The final expressions for the coefficients of the expansion series of solutions have the form of convolutions in time. The cores of these integral representations are the original coefficients of the series for the influence functions. As an example, the response of a moment elastic rod to the action of a non-stationary axial load is considered. The results obtained are illustrated graphically. The practical convergence of expansion series is estimated.

— Using generalized functions, Green’s functions for homogeneous elastic isotropic half-planes and half-spaces are constructed. Airy and Maxwell stress functions are used to find Green’s functions. One-dimensional and two-dimensional integral Fourier transforms are used to solve the boundary value problems. Taking into account the properties of generalized functions with a point support, singular components of displacement images are distinguished. It is shown that they correspond to the rigid-body displacement. If there are no singular components, then the stresses and di-splacements coincide with the known classical solutions of the Flamant, Boussinesq, and Cerutti problems.

The problem of unsteady bending of an elastic diffusion orthotropic cantilever Timoshenko beam under loading applied to its free end is considered. The model takes into account that the velocity of propagation of diffusion perturbations is finite due to diffusion flux relaxation. The elastic diffusion processes are described by a coupled system of equations for the Timoshenko beam with allowance for diffusion. A solution of the problem is sought by the method of equivalent boundary conditions. For this purpose, an auxiliary problem is considered, whose solution is obtained by applying the Laplace integral transform in time and trigonometric Fourier series expansions in space. Next, relations connecting the right-hand sides of the boundary conditions of the original and auxiliary problems are constructed. These relations represent a system of Volterra integral equations of the first kind. The system is solved numerically by applying quadrature rules. For an orthotropic beam made of a three-component material, the interaction of unsteady mechanical and diffusion fields is numerically analyzed. Finally, the main conclusions concerning the coupling effect of the fields on the stress-strain state and mass transfer in the beam are given.

We consider the problem of determining the stress-strain state of an orthotropic multicomponent cylinder affected by unsteady surface elastic diffusive perturbations. The coupled system of elastic diffusion equations in the polar coordinate system is used as a mathematical model. Diffusion relaxation effects, implying finite rates of diffusion flux propagation, are taken into account. The solution to this problem is sought in the integral form and is represented as convolutions of Green’s functions with functions defining surface elastodiffusive perturbations. We use the Laplace transform by time and Fourier series expansion in Bessel functions of the first kind to find Green’s functions. The Laplace transform inversion is done analytically due to residues and operational calculus tables. An analytical solution to the problem is obtained. A numerical study of the interaction of mechanical and diffusion fields in a continuous orthotropic cylinder is performed. We used three-component material as an example. The cylinder is under pressure, which is uniformly distributed over it surface. We use three-component material as an example.

A polar-symmetric elastic diffusion problem is considered for an orthotropic multicomponent homogeneous cylinder under uniformly distributed radial unsteady volumetric perturbations. Coupled elastic diffusion equations in a cylindrical coordinate system is used as a mathematical model. The model takes into account a relaxation of diffusion effects implying finite propagation speed of diffusion perturbations. The solution of the problem is obtained in the integral convolution form of Green's functions with functions specifying volumetric perturbations. The integral Laplace transform in time and the expansion into the Fourier series by the special Bessel functions are used to find the Green's functions. The theory of residues and tables of operational calculus are used for inverse Laplace transform. A calculus example based on a three-component material, in which two components are independent, is considered. The study of the mechanical and diffusion fields interaction in a solid orthotropic cylinder is carried out.

We consider an unsteady elastic diffusion vibration problem of an orthotropic Timoshenko beam on an elastic foundation under a distributed transverse load. The Winkler model is used as an elastic foundation model. We use the system of Timoshenko beam bending equations taking into account diffusion for the mathematical problem formulation. These equations are obtained with the d’Alembert variational principle applied to the elastic diffusion continuum model. The resulting model considers the diffusion fluxes relaxation. The problem solution is sought as convolutions of Green’s functions with functions defining unsteady distributed disturbances. The integral Laplace transform in time and the expansion in the Fourier series in the longitudinal coordinate are used to find the Green’s functions. A calculation example for the beam with a rectangular section is considered. The beam deflections and the concentration increments under the action of an impulsively applied distributed transverse load are found. Finally, the main conclusions about the coupling effect of the stress–strain state and mass transfer in the beam are represented.

The initial stage of indentation of a convex rigid punch into isotropic elastic half-plane with friction is considered. A closed mathematical formulation in Cartesian coordinates is presented by equations of motion in the absence of a body force for homogeneous isotropic elastic medium in terms of displacement potentials, Cauchy equations for deformations, Hooke law and translational equation for the punch. Initial conditions are homogeneous. Outside of contact region surface is free from stresses. Inside contact region normal displacements of the surface of a half-space and the surface of a punch are taken to be equal and the relation between tangential and normal stresses in a form of Coulomb friction law is given. Due to the short duration of the supersonic stage and smallness of the contact region radius the tangential stress direction is considered to be point independent. Resolving functional equations are given in a form of convolutions with influence functions. The latter is the solution the original problem for half-plane with a special boundary condition in a form of the Dirac delta function. This solution is obtained in the domain of Laplace transform by time and Fourier transform by spatial coordinate. The original is constructed with the combined Fourier–Laplace inversion method. Solution is given with consideration to the initial stage characteristic property of supersonic velocity of contact region expansion (no smaller than propagation velocity of expansion-compression waves). It is shown that kinematic parameters of the punch, resulting force and discontinuities of the first kind on boundaries of the contact region are independent of friction unlike contact pressure. Calculation examples for different values of friction coefficient are presented.

We investigated unsteady elastic diffusion vibrations of a rectangular isotropic Timoshenko plate. For the mathematical problem formulation, a model of coupled elastic diffusion processes in a multicomponent continuum is used. Using the d'Alembert variational principle, the equations of transverse vibrations of a rectangular isotropic Timoshenko plate taking into account diffusion are obtained from this model. An initialboundary value problem of a simply supported plate bending is formulated.

We study unsteady elastic diffusion vibrations of a freely supported rectangular isotropic Kirchhoff-Love plate on an elastic foundation, which is under the action of a distributed transverse load. A model that describes coupled elastic diffusion processes in multicomponent continuum is used for the mathematical problem formulation. The longitudinal and transverse vibrations equations of a rectangular isotropic Kirchhoff-Love plate with diffusion were obtained from the model using the d'Alembert variational principle. The problem solution of unsteady elastic diffusion plate vibrations is sought in integral form. The bulk Green's functions are the kernels of the integral representations. To find the Green's functions, we used the Laplace transform in time and the expansion into double trigonometric Fourier series in spatial coordinates. Green's functions in the image domain are represented in the form of rational functions depends on the Laplace transform parameter. The transition to the original domain is done analytically through residues and tables of operational calculus. The bulk Green's functions analytical expressions are obtained. Using a two-component continuum, a numerical study of unsteady mechanical and diffusion fields interaction is done for an isotropic plate. The solution is presented in analytical form, as well as in the form of three-dimensional graphs of the displacement fields and concentration increments on time and coordinates.

A spatial transient contact problem with moving boundaries for a thin elastic cylindrical shell and a rigid indenter bounded by a smooth convex surface is considered. A closed mathematical formulation is given and a system of resolving equations is constructed. The main integral equation follows from the principle of superposition and contact conditions. The core of this equation is the transient function for the cylindrical shell. To a closed system of resolving equations, it is supplemented by a kinematic relation for determining the moving boundary of the contact region and the equation of motion of the indenter as a rigid body. An algorithm for solving the spatial non-stationary contact problem for an infinitely long cylindrical shell and rigid indenter in the case of a normal impact on the side surface of the shell is constructed and implemented. Examples of calculations are given.

We considered the problem of unsteady direct bending of an isotropic homogeneous elastodiffusive cantilevered Euler-Bernoulli beam. For the mathematical formulation of the problem, we use the closed system of equations of transverse unsteady beam vibrations with inner diffusion. The formulation is based on the model of elastic diffusion using the d'Alember variational principle. It is assumed that the deflections of the beam are small and the hypothesis of flat sections is fulfilled. The Euler-Bernoulli hypothesis is valid for section rotations. A solution to the problem is sought using the method of equivalent boundary conditions, which allows a transition from the initial formulation with arbitrary boundary conditions to the problem of the same type and with the same domain geometry. First, an auxiliary problem is solved using the integral Laplace transform in time and trigonometric Fourier series. Then, some relations connecting the boundary conditions right-hand sides of the original and auxiliary problems are constructed. These relations are the Volterra integral equations of the first kind. For solving this system, quadrature formulas of an average rectangle are used. Finally, the solution of the original problem is represented in the form of the convolution of the Green's functions for the auxiliary problem with the functions determined by solving the system of the Volterra integral equations. The interaction between unsteady mechanical and diffusion fields is analyzed using an isotropic beam as an example. Graphs showing the dependence of the displacement fields and concentration increments on time and coordinates are given. Analysis of the results obtained led to conclusion that the coupling action of mechanical and diffusion fields affects the stress-strain state and mass transfer in a beam

Рассматривается полярно-симметричная задача механодиффузии для ортотропного сплошного многокомпонентного цилиндра, находящегося под действием равномерно распределенного по поверхности внешнего давления. Приложенные нагрузки инициируют массоперенос, который в свою очередь влияет на напряженно-деформированное состояние цилиндра. В качестве математической модели используется связанная система дифференциальных уравнений упругой диффузии в цилиндрической системе координат, которая учитывает релаксационные диффузионные эффекты, подразумевающие конечные скорости распространения диффузионных потоков. Задача решается с помощью метода эквивалентных граничных условий, заключающегося в том, что вначале рассматривается некоторая вспомогательная задача, решение которой известно и отличающаяся от исходной задачи только граничными условиями. Затем строится соотношение, связывающее правые части граничных условий обеих задач. Указанное соотношение представляет собой интегральное уравнение, решение которого ищется с помощью квадратурных формул. Из этого уравнения находятся правые части граничных условий вспомогательной задачи. В результате решение исходной задачи находится в виде сверток функций Грина вспомогательной задачи с функциями, полученными при решении вышеуказанного интегрального уравнения. Метод эквивалентных граничных условий разработан для начально-краевых задач, решение которых невозможно получить методом разделения переменных. Для нестационарных задач он является полуаналитическим, в стационарных и статических задачах он позволяет получить решение в аналитической форме. На примере трехкомпонентного материала выполнено исследование взаимодействия механического и диффузионного полей в сплошном ортотропном цилиндре. Исследованы предельные переходы к статическим механодиффузионным режимам, а также к классическим моделям упругости. Промоделировано влияние релаксационных эффектов на кинетику массопереноса в сплошных средах. Результаты исследований представлены в аналитической и графической формах.

We investigated an unsteady elastic diffusion vibration of a simply supported rectangular isotropic Kirchhoff-Love plate. The plate is under the action of a distributed transverse load. A model that describes coupled elastic diffusion processes in a multicomponent continuum is used for the mathematical problem formulation. The model is taking into account the diffusion fluxes relaxation. The transverse vibration equations of a rectangular isotropic Kirchhoff-Love plate with diffusion were obtained from the model using the d'Alembert variational principle. The initial-boundary value problem of a freely supported isotropic rectangular plate bending is formulated on the basis of the obtained equations. The plate is under the action of elastic diffusion perturbations distributed over the surface. The problem solution of an unsteady elastic diffusion plate vibration is sought in an integral form. The surface Green's functions are the kernels of the integral representations. To find the Green's functions, we used the Laplace transform in time and the expansion into double trigonometric Fourier series in spatial coordinates. Green's functions in the image domain are represented in the form of rational functions and depend on the Laplace transform parameter. The transition to the original domain is done analytically through residues and tables of operational calculus. The surface Green's function analytical expressions are obtained. As a calculation example, we considered a freely supported elastodiffusive plate under the action of suddenly applied unsteady bending moments distributed over the plate surface. By using a three-component continuum, a numerical study of interactions between unsteady mechanical and diffusion fieldsis done for an isotropic plate. The influence of relaxation effects on the kinetics of mass transfer is investigated. The solution is presented in the analytical form, as well as in the graphs of the displacement fields and concentration increments on time and coordinates. At the end of the publication, the main conclusions are given about the fields coupling effect and the relaxation of diffusion fluxes on the stress-strain state and mass transfer in the plate.

Рассматривается нестационарная задача о нестационарном изгибе однородной изотропной пластины Тимошенко с учетом диффузии. Исходная математическая постановка включает в себя систему уравнений нестационарных изгибных колебаний пластины с учетом диффузии, которая получена из общей модели механодиффузии для сплошных сред с помощью вариационного принципа Даламбера. Сформулирована начально-краевая задача о нестационарном изгибе свободно опертой прямоугольной пластины. Проверены предельные переходы к модели Кирхгофа – Лява.