Volume 236, Nº 2 (2019)

We consider specific realizations of different implicit generalized Runge–Kutta methods as applied to the numerical integration with respect to time of initial-boundary-value problems for the second-order parabolic equations and investigate their spectral stability. It is shown that all implicit generalized Runge–Kutta methods are unconditionally spectrally stable but some of them have the conditional property of monotonicity of the numerical solution with respect to time. The functions of spectral stability of the implicit generalized Runge–Kutta methods are rational. We compare the analytic solution of the nonstationary one-dimensional problem of heat conduction with the numerical solutions of this problem obtained by different implicit generalized Runge–Kutta methods. It is shown that, in this case, the application of the one-stage Radau methods with subsequent discretization of the problem with respect to the space variable leads to the classical forward finite difference scheme (Laasonen scheme), whereas the use of the one-stage Gauss–Legendre method leads to a six-point symmetric scheme (Crank–Nicolson scheme). It is shown that diagonally implicit generalized Nørsett and Burrage methods are realized in almost the same way as the one-stage Radau and Gauss–Legendre methods but their accuracy in the time step is 10–1000 times higher. On the basis of comparison of the numerical and analytic solutions, we conclude that, in order to get practically suitable numerical solutions without any restrictions on the time step, it is reasonable to use one- and three-stage generalized Radau methods or two- and four-stage Lobatto IIIC methods. All other explicit and implicit generalized Runge–Kutta methods require certain restrictions imposed on the time step.

We compute the effective thermal contact resistance of a pair of materials formed by AISI 304 stainless steel and 200 nickel alloy in the presence of a periodic array of gaps filled with a heat-conducting medium (argon, water, helium, oxygen, or air) on the interface of these materials. In the nondeformed state, the surface of AISI 304 stainless steel has a microtextured topography formed by regularly located grooves of identical shapes, while the surface of 200 nickel alloy is perfectly flat. We estimate the level of thermal rectification for structures of this kind for different values of the applied mechanical and thermal loads and the maximal height of the surface grooves.

A micromechanical model is developed to determine the effective inelastic properties of nanocomposites under monoharmonic deformation by taking into account the detailed microstructural geometries and constitutive models of the constituents. By using the correspondence principle in viscoelasticity and the modified Mori–Tanaka method, the effects of interface between the inclusion and the matrix are taken into account. By applying the developed model, we perform the numerical analysis aimed at the determination of complex moduli for a polymeric nanocomposite reinforced with nanofibers composed from carbon nanotubes under the isothermal conditions. The dependences of complex moduli on temperature and the amplitude of strain intensity are analyzed. The composites reinforced with unidirectionally aligned nanofibers are considered. The accumulated results demonstrate a strong dependence of the storage and loss moduli on temperature within a broad temperature range. The storage and loss moduli are found to monotonically increase with the volume fraction of nanofibers. At the same time, they decrease as temperature increases. The obtained results show that the strength of material decreases as temperature increases in the elastic and inelastic regions and the inelastic behavior occurs for lower strain amplitude as temperature increases.

We formulate mathematical models for the evaluation of the lifetime of plates with systems of cracks under the action of long-term force loads in the presence of high temperatures and corrosive media. These models are based on the use of the well-known main mechanisms of propagation of creep cracks, corrosion fracture, and the first law of thermodynamics (balance of energy components and rates of their changes) for a metal plate containing a system of macrocracks and subjected to the action of long-term tension and corrosive media at high temperatures. We consider the case of a double-periodic system of cracks.