Technique for selecting the functions of the constitutive equations of creep and long-term strength with one scalar damage parameter

The strain-strength characteristics of aerostructures made of hardening materials under uniaxial tension in creep conditions are determined. The problem is reduced to a system of ordinary differential equations of the kinetic theory of creep with one scalar damage parameter. The approximate solutions of the problem are obtained with the help of the implicit Euler method and of the arc length method in combination with the explicit methods of the Runge–Kutta family for cylindrical St.45 steel samples and 3V titanium alloy plates.     

Stochastic model of nonisothermal creep and long-term strength of materials

A stochastic model of nonisothermal creep and long-term strength of metallic materials is proposed. Experimental data on the creep of the ZhS6KP alloy at temperatures equal to 900, 950, and 1000°C are stochastically analyzed. These experimental data are used to substantiate the hypotheses applied in constructing the model. The stochastic model is checked for adequacy to the experimental data on the creep of the ZhS6KP alloy under stationary and nonstationary loading conditions. It is shown that the calculated and experimental data are in satisfactory agreement.     

Long-term strength of metals and creep equations based on the Coulomb-Mohr criterion

It is proposed to construct long-term strength and creep relations for metals on the basis of the Coulomb-Mohr criterion. The creep equations and the long-term strength criterion for plane stress are analyzed in detail. Results of long-term strength calculations are compared with data of experiments with metallic materials. It is established that theoretical and experimental results are in satisfactory agreement. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 115–123, November–December, 2007.     

Justification of the energy variant of the theory of creep and long-term strength of metals

The hypotheses underlying the energy variant of the theory of creep and long-term strength of metals are formulated and justified experimentally.     

Stochastic differential equations for the kinetic and hydrodynamic equations

A finite system of stochastic interacting particles is considered. The system approximates the solutions of the kinetic equations (the Boltzmann equation, the Boltzmann-Enskog equation) as well as the solutions describing the macroscopic evolution of fluids: the Euler and the Navier-Stokes hydrodynamic equations.     

Fracture criteria under creep with strain history taken into account,and long-term strength modelling

In the present paper, we continue to study the nonlinear constitutive relation (CR) between the stress and strain proposed in [1] to describe one-dimensional isothermal rheological processes in the case of monotone variation of the strain (in particular, relaxation, creep, plasticity, and superplasticity). We show that this CR together with the strain fracture criterion (FC) leads to theoretical long-term strength curves (LSC) with the same qualitative properties as the typical experimental LSC of viscoelastoplastic materials. We propose two parametric families of fracture criteria in the case of monotone uniaxial strain, which are related to the strain fracture criterion (SFC) but take into account the strain increase history and the dependence of the critical strain on the stress. Instead of the current strain, they use other measures of damage related to the strain history by time-dependent integral operators. For any values of the material parameters, analytic studies of these criteria allowed us to find several useful properties, which confirm that they can be used to describe the creep fracture of different materials. In particular, we prove that, together with the proposed constitutive relations, these FC lead to theoretical long-term strength curves (TLSC) with the same qualitative properties as the experimental LSC. It is important that each of the constructed families of FC forms a monotone and continuous scale of criteria (monotonously and continuously depending on a real parameter) that contains the SFC as the limit case. Moreover, the criteria in the first family always provide the fracture time greater than that given by the SFC, the criteria in the second family always provide a smaller fracture time, and the difference can be made arbitrarily small by choosing the values of the control parameter near the scale end. This property is very useful in finding a more accurate adjustment of the model to the existing experimental data describing the fracture time dependence on the stress, temperature, radiation, and other factors: if these data are poorly described by the SFC, then one can choose a more appropriate criterion from the constructed families by varying the value of the control parameter smoothly and monotonously.     

Accuracy of model kinetic equations

The accuracy of model kinetic equations is analyzed using the exact moment solutions of the Boltzmann–Maxwell equation for homoenergetic affine flows of a monatomic gas of Maxwellian molecules in the absence of external forces. Solutions of the third-order kinetic-moment equations for homogeneous shear flow and one-dimensional homogeneous expansion-collapse flow are considered. The principal advantages of the domestic Shakhov and, especially, Larina–Rykov models are demonstrated.     

Group-invariant solutions of kinetic equations

In the development of analytic methods of solution of kinetic equations, it is expedient to use group raetliods. The establishment of a symmetry group makes it possible to justify the choice of a definite model of kinetic equation corresponding to the physical formulation of the problem, to solve the Cauchy problem in a number of cases, and to obtain classes of new exact solutions that can be used as standards in the construction of numerical algorithms for solving kinetic equations. Bobylev [1–4] and Krook and Wu [5, 6] used group methods to analyze the spatially homogeneous Boltzmann equation in the case of isotropy with respect to the velocities and Maxwellian molecules. They obtained exact solutions and investigated the asymptotic behavior of the main equation. In the present paper, group methods are used to find and analyze exact solutions of the Bhatnagar-Gross-Krook kinetic equation, which successfully simulates the basic properties of the Boltzmann equation. Conclusions are drawn about the symmetries of the Boltzmann equation. To simplify the calculations, the exposition is presented for the case of the one-dimensional Bhatnagar-Gross-Krook equation with constant effective collision frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 135–140, July–August, 1982.     

Some types of kinetic equations reducible to partial differential equations

The possibility of passing from the kinetic equation to a partial differential equations is rigorously mathematically proved for the case of nearly elastic scattering processes. Some examples are considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 12–16, September–October, 2007.