Contact problems of thermoelasticity taking account of heat production

We give a survey of papers that discuss the mathematical statement and methods of solution of contact problems of thermoelasticity taking account of heat production due to frictional forces on the surface of contact.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 93–100.     

Contact problems for functionally graded materials of complicated structure

An approximate analytical method allowing one to efficiently solve, to a preassigned accuracy, contact problems for materials with properties arbitrarily varying in depth is developed. Its possibilities are illustrated with the example of torsion of an elastic half-space, having a coating inhomogeneous across its thickness, by a circular stamp. All the results obtained are rigorously substantiated. For the approximate solutions constructed, their error is analyzed. The asymptotic properties of the solutions are investigated. The cases of a nonmonotonic change in the elastic properties are considered. In particular, the analytical solutions are examined in the case where the variation gradient of the elastic properties changes its sign many times. The results derived allow one to solve the inverse problems of elasticity theory of inhomogeneous media (e.g., the problem on controlling the variation in the elastic properties of a covering across its thickness).     

Strain gradient solutions of half-space and half-plane contact problems

General solutions for the problems of an elastic half-space and an elastic half-plane, respectively, subjected to a symmetrically distributed normal force of arbitrary profile are analytically derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. Mindlin’s potential function method and Fourier transforms are employed in the formulation, and the half-space and half-plane contact problems are solved in a unified manner. The specific solutions for the problems of a half-space/plane subjected to a concentrated normal force or a uniformly distributed normal force are obtained by directly applying the general solutions, which recover the existing classical elasticity-based solutions of the Flamant and Boussinesq problems as special cases. In addition, the indentation problems of an elastic half-space indented by a flat-ended cylindrical punch, a spherical punch, and a conical punch, respectively, are solved using the general solutions, leading to hardness formulas that are indentation size- and material microstructure-dependent. Numerical results reveal that the displacement and stress fields in a half-space/plane given by the current SSGET-based solutions are smoother than those predicted by the classical elasticity-based solutions and do not exhibit the discontinuity and/or singularity displayed by the latter. Also, the indentation hardness values based on the newly obtained half-space solution are found to increase with decreasing indentation radius and increasing material length scale parameter, thereby explaining the microstructure-dependent indentation size effect.     

Thermal shock fracture of a crack in a functionally gradient half-space based on the memory-dependent heat conduction model

In the present study, we consider a thermoelastic half-space made of a functionally gradient material with an insulated crack, which is subjected to a thermal impact. The memory-dependent heat conduction model is adopted for analysis. By using the Fourier and Laplace transforms, the thermoelastic problem is formulated in terms of singular integral equations which can be solved numerically. Effects of the time delay, kernel function, and nonhomogeneity parameters on the temperature and stress intensity factor are analyzed. Our results are also compared with those based on the Fourier and CV heat conduction models, which can be viewed as two special cases of the present model. In conclusion, the memory-dependent derivative and nonhomogeneity parameters play an essential role in controlling the heat transfer process.     

Some quasi-static problems for a viscoelastic half-space

Some problems for a viscoelastic half-space are solved in the case of noncommutative operators. A solution of the equilibrium equation analogous to the Boussinesq-Papkovich solution is constructed. The problem of a normal pressure acting on the boundary of a viscoelastic half-space is solved. Two forms of this solution are obtained and both are used in the following problems, the problem of a concentrated load moving over the boundary of a half-space and the problem of a circular rigid stamp. The case of periodic motion of a periodic load is investigated with reference to the example of motion in a circle. At constant Poisson's ratio the solution of the problem of a stamp can be used for determining the creep or relaxation function.Mekhanika Polimerov, Vol. 2, No. 3, pp. 392–402, 1966Presented 12 November 1965 at the Riga Conference on Polymer Mechanics.     

Stability of a three-layer operator-difference scheme for coupled thermoelasticity problems

Stability of a three-layer operator-difference scheme with weights, which generalizes a class of difference and projection-difference schemes for linear coupled thermoelasticity problems, is analyzed. Energy estimates for the solution and its first-order grid derivative are obtained.     

Splitting scheme for poroelasticity and thermoelasticity problems

Boundary value problems in thermoelasticity and poroelasticity (filtration consolidation) are solved numerically. The underlying system of equations consists of the Lamé stationary equations for displacements and nonstationary equations for temperature or pressure in the porous medium. The numerical algorithm is based on a finite-element approximation in space. Standard stability conditions are formulated for two-level schemes with weights. Such schemes are numerically implemented by solving a system of coupled equations for displacements and temperature (pressure). Splitting schemes with respect to physical processes are constructed, in which the transition to a new time level is associated with solving separate elliptic problems for the desired displacements and temperature (pressure). Unconditionally stable additive schemes are constructed by choosing a weight of a three-level scheme.     

Stability of an operator-difference scheme for thermoelasticity problems

We study a linear three-layer operator-difference scheme with weights which generalizes a class of difference and projection-difference schemes for coupled thermoelasticity problems. Using the method of energy inequalities, we obtain stability estimates in grid energy norms under certain conditions on operator coefficients and parameters of the scheme.     

Three-dimensional mixed problems of thermoelasticity for isotropic plates

We obtain the homogeneous thermal solutions due to a temperature field for the three-dimensional thermoelastic problem for isotropic plates on whose plane faces homogeneous thermal and mixed mechanical conditions of flat face and diaphragm type are prescribed. This makes it possible to reduce the thermoelastic boundary-value problem to the corresponding elasticity problem. Translated fromTeoreticheskaya i Prikladnaya Mekhanika. No. 25, 1995, pp. 3–8.     

Solution of two-dimensional problems of elasticity and thermoelasticity for a rectangular region

We propose a separation-of-variables method for the biharmonic equation and construct a complete system of orthogonal functions for constructing exact solutions in the form of non-periodic trigonometric series for two-dimensional problems of the theory of elasticity and thermoelasticity for a rectangular region. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 19–25.     

Semiconcavity results for constrained optimal control problems in a half-space

In this paper we give semiconcavity results for the value function of some constrained optimal control problems with infinite horizon in a half-space. In particular, we assume that the control space is the l¹-ball or the l^∞-ball in Rn.     

Initial-boundary value problems for the Vlasov-Poisson equations in a half-space

We consider initial-boundary value problems for the Vlasov-Poisson equations in a half-space that describe evolution of densities for ions and electrons in a rarefied plasma. For sufficiently small initial densities with compact supports and large strength of an external magnetic field, we prove the existence and uniqueness of classical solutions for initial-boundary value problems with different boundary conditions for the electric potential: the Dirichlet conditions, the Neumann conditions, and nonlocal conditions.     

Contact problem for elastic half-space with an ellipsoidal cavity

We consider the axisymmetric problem of elasticity theory for a space with an elongated ellipsoidal cavity with mixed boundary conditions of smooth contact on its surface. The method of p-analytical functions is applied to reduce the solution of the problem to an infinite quasi-completely regular system of linear algebraic equations with upper bounded free terms that tend to zero as the index increases. The behavior of the normal stress near the contact line of the different boundary conditions is analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 94–103, 1988.     

Acceleration of partitioned coupling schemes for problems of thermoelasticity

A partitioned coupling scheme for problems of thermo-elasticity at finite strains is presented. The coupling between the mechanical and thermal field is one of the most important multi-physics problem. Typically two different strategies are used to find an accurate solution for both fields: Partitioned or staggered coupling schemes, in which the mechanics and heat transfer is treated as a single field problem, or a monolithic solution of the full problem. Monolithic formulations have the drawback of a non-symmetric system which may lead to extremely large computational costs. Because partitioned schemes avoid this problem and allow for numerical formulations which are more flexible, we consider a staggered coupling algorithm which decouples the mechanical and the thermal field into partitioned symmetric sub-problems by means of an isothermal operator-split. In order to stabilize and to accelerate the convergence of the partitioned scheme, two different methods are employed: dynamic relaxation and a reduced order model quasi-Newton method. A numerical simulation of a quasi-static problem is presented investigating the performance of accelerated coupling schemes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On continuous dependence for a class of dynamical problems in linear thermoelasticity

Given that the temperature dependent elasticities of a linear anisotropic elastic material satisfy certain Lipschitz type conditions, it is shown that the displacement vector depends continuously, in an appropriate norm, on temperature deviations from a fixed but arbitrary steady state temperature distribution.     

Green functions of three-dimensional static thermoelasticity problems for a piecewise homogeneous space

The Green functions of thermoelasticity problems for a piecewise homogeneous body, composed of two perfectly contacting semiinfinite isotropic bodies, have been constructed in closed form. We have used here generalized functions and Green functions for the corresponding systems of ordinary differential equations. As the limiting cases, we have obtained the Green functions of thermoelasticity problems for a semiinfinite body, when its surface, thermally insulated or maintained at zero temperature, is load-free or rigidly restrained. We also present some results of numerical studies.