Boundary-value contact problem of thermoelastoplasticity for a two-layer eccentric cylindrical pipe

We formulate the plane two-dimensional static boundary-value contact problem of thermoelastoplasticity for a two-layer eccentric cylindrical pipe under the action of a temperature field and compressive normal stresses that are uniformly distributed on its lateral surfaces and present its approximate solution. We assume that the mechanical and thermophysical properties of the materials are temperature-independent, plastic strains arise on the interior lateral surface of the two-layer pipe and completely envelop it, and the material of the pipe is perfectly elastoplastic, incompressible in the domain of plasticity, and satisfies the Tresca-Saint-Venant plasticity condition. I. Franko L'viv State University, L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 57–66, April–June, 1998.     

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.     

Dynamic problem of thermoelasticity for a two-layer space

The method of influence functions is applied to construct a closed-form solution of the dynamic problem of thermoelasticity for a two-layer space (in one-dimensional framework) with nonideal thermal contact and ideal mechanical contact at the layer interface. The influence functions are constructed by the operational method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 95–102, 1985.     

The antiplane problem of the motion of a point shear load over a two-layer elastic base at a constant velocity

The problem of modelling the motion of a force disturbance in an elastic medium that is heterogeneous over its depth is investigated. It is in an antiplane formulation in a moving system of coordinates that all possible versions of the ratio of the velocity of motion of the surface point shear load to the velocities of the shear waves in the layers of the two-layer elastic base are examined. Cases of a subsonic regime (SBR) in the upper and lower layers, of a supersonic regime (SPR) in the upper layer and an SBR in the lower layer, and of an SBR in the upper layer and an SPR in the lower layer are studied using the Fourier transform and the theory of residues. The last two cases are extremely interesting from the mathematical point of view, as here, on the boundary between the layers, the solutions of elliptic and hyperbolic equations meet, and previously unknown features arise in the displacements that,it seems, should also occur in the solution of the corresponding plane problem. The case of an SPR in the upper and lower layers is investigated using a special method for successive allowance for the incident, reflected and refracted shock wave fronts. In all cases, expressions are obtained for the displacements in the layers, and their characteristic features are investigated.     

The thermoelastic problem for a spherical shell with slits

We study the problem for a spherical shell weakened by thermally insulated slits. Using the two-dimensional Fourier transform and the elements of the theory of distributions, we obtain the resolvent systems of singular integral equations. We carry out numerical studies of the influence of the geometric parameters of the shell and the slits on the values of the coefficients of intensity of the forces and moments. Three figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 75–81.     

Generalized dynamic problem of thermoelasticity for a two-layer space

A closed-form solution of the generalized unsteady heat-conduction problem and dynamic thermoelasticity problem is constructed for an elastic isotropic two-layer space free from external disturbances. Some common practical cases are examined.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 56–64, 1987.     

Free boundary problem for a two-layer inviscid incompressible fluid

The existence of a local (in time) classical solution of a free boundary problem for a two-layer inviscid incompressible fluid is shown. The method of successive approximations and the novel approach to Lagrangian coordinates of Solonnikov are used.     

The three-dimensional contact problem for a wedge-shaped valve

The three-dimensional contact problem for an elastic wedge-shaped valve, situated in a wedge-shaped cavity in an elastic space, is investigated. A regular asymptotic method is used to solve the integral equation of this problem. The method is effective for a contact area relatively far from the edge of the wedge-shaped cavity. Calculations are carried out. The solutions of the three-dimensional auxiliary problems on the equilibrium of an elastic wedge-shaped cavity and an elastic wedge are based on well-known Green's functions, constructed using Fourier and Kontorovich–Lebedev integral transformations.     

Dynamics of a two-layer fluid sloshing problem

A second-order ordinary differential equation, which is a reducedform of the periodically forced extended Korteweg–de Vries(eKdV) equation, is derived in the physical context of sloshinga two-layer fluid tank. In the limit of small dispersion, numericalevidence is given of multiple periodic solutions displayingfast oscillations superimposed on slow periodic waves and ahigher-order Melnikov method is then used to verify the existenceof such solutions. The dynamical behaviour of a similar equationwith more general coefficients is also examined, demonstratingthe existence of periodic and chaotic behaviour. We highlightnew aspects which arise due to the presence of mixed nonlinearity.     

The contact problem for a geometrically non-linear Timoshenko-type shell

An algorithm is developed for the numerical solution of the contact problem of an elastic Timoshenko-type shell subjected to arbitrarily large displacements and rotations, using mixed finite-element approximations. It is essential that six displacements of the faces of the shell are chosen as the required functions. This enables one, first, to simplify the formulation of contact problems in the mechanics of thin-walled structures, since functions by means of which the conditions for the non-penetration of the bodies are formulated are chosen as the required functions and, second, to obtain relations for the components of the Green-Lagrange strain tensor in curvilinear, orthogonal coordinates which accurately represent arbitrarily large displacements of a shell as a rigid body.     

The nonsteady-state planar oscillations of a plate and oscillator on a two-layer base

We consider a plane dynamic contact problem for an inhomogeneous base of the following form: a soft layer on a rigid layer of an elastic half-plane. The layer is represented by a Winkler model, corresponding to the long-wave asymptotics of the equations of elasticity theory. The problem is reduced to a system of integro-differential equations that is solved numerically. We present the results of the computations of dynamic characteristics describing the oscillations of a rigid body and an oscillator on this base.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 64–68.     

The nonstationary heat-conduction problem for a heat-sensitive space with a spherical cavity

We propose a method of finding closed-form solutions of nonstationary heat-conduction problems for heat-sensitive spaces with simple nonlinearity. The approach is illustrated using the example of a space with a spherical cavity. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 100–104.     

The contact problem for a piecewise-homogeneous plane with a finite inclusion

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite inclusion, which meets the interface at a right angle and is loaded with shear forces, is considered. The contact stresses along the contact line are determined, and the behaviour of the contact stresses in the neighbourhood of singular points is established. By using the methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite section. Using an integral transformation, a Riemann problem is obtained, the solution of which is presented in explicit form.     

The contact problem for a piecewise-homogeneous plane with a semi-infinite inclusion

A piecewise-homogenous elastic plate, reinforced with a semi-infinite inclusion, which intersects the interface at a right angle and is loaded with shear forces is considered. The contact stresses along the contact line are determined and the behaviour of the contact stresses in the neighbourhood of singular points is established. Using methods of the theory of analytical functions and integral transformations the problem is reduced to a system of singular integro-differential equations on the semi-axis. The solution is presented in explicit form.     

The reissner-sagoci problem for a multilayer base with a cylindrical cavity

We study the problem of the torsional oscillations of a plane disk-shaped die coupled with the upper boundary of a multilayer elastic base containing a vertical cylindrical cavity whose axis is perpendicular to the interface of the layers. The problem is stated as paired integral equations connected with the Weber integral transforms. To couple the solutions in the layers we use the method of initial parameters, which makes it possible to express the stress-strain state in any layer in terms of the solution of a Fredholm integral equation of second kind, to which the paired equations reduce. We exhibit an algorithm for numerical implementation of the problem. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 55–61.     

The solution of the contact problem for a circular plate

An approach, based on Rektorys’ theorem on the minimum of a quadratic functional which, without any fundamental difficulties, can be used for diverse contact problems, is used to solve the problem of the contact interaction of a circular flexible plate with an elastic half-space.     

The contact problem for a rectangle with stress-free side faces

The plane contact problem for an elastic rectangle into which two symmetrically positioned punches are impressed is considered. Homogeneous solutions are constructed that leave the side faces of the rectangle stress-free. When the modified boundary conditions using generalized orthogonality of the homogeneous solutions are satisfied, the problem reduces to a Friedholm integral equation of the first kind in the function describing the displacement of the surface of the rectangle outside the contact area. This function is sought in the form of the sum of a trigonometric series and a power function with a root singularity. The ill-posed infinite system of algebraic equations thereby obtained is regularized by introducing a small positive parameter (Ref. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978), and, after reduction, has a stable regularized solution. Since the matrix elements of the system are determined by a poorly converging number series, an effective method was developed for calculating the residues of the series. Formulae are found for the contact pressure distribution function and dimensionless indentation force. Since the first formula contains a third-order derivative of the functional series, when it is used, a numerical differentiation procedure is employed (Refs. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978; Danilina NI, Dubrovskaya NS, Kvasha OP et al. Numerical Methods. Textbook for Special Colleges. Moscow: Vysshaya Shkola; 1976). Examples of a calculation for a plane punch are given.     

The contact problem of electroelasticity for a plane elliptic die

We obtain an exact solution of the problem of the stress-strain state of an elastic piezoelectric half-space acted on by a rigid elliptic die with a flat base. The axis of symmetry of the body coincides with the direction of the field of preliminary polarization of the body. The solution is confined to the case of translational displacement of the die. We determine the quantities that characterize the mechanical and electric fields that arise in the region of contact of the die with the half-space. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 40–52.